Symbolic Logic [5 ed.] 0023249803, 9780023249808

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SYMBOLIC LOGIC

SYMBOLIC FIFTH

EDITION

IRVING M. COPI

Macmillan Publishing NEW YORK Collier

Co., Inc.

Macmillan Publishers LONDON

University of Hawaii

Copyright

©

1979, Irving

M. Copi

Printed in the United States of America All rights reserved.

No

book may be reproduced or transmitted

part of this

in

any

form or by any means, electronic or mechanical, including photocopying, recording, or any information storage

and

retrieval system, without permission in writing

the Publisher.

and copyright

Earlier editions copyright 1954

1965 by Macmillan Publishing Co., copyright

©

©

Inc.,

1967 and 1973 by Irving M. Copi

Macmillan Publishing Co., Inc. 866 Third Avenue, New York, New York 10022 Collier

Macmillan Canada, Ltd.

Library of Congress Cataloging in Publication Data Copi, Irving M.

Symbolic

logic.

Includes index. 1.

Logic, Symbolic and Mathematical.

BC135.C58 1979 ISBN 0-02-324980-3 Printing:

1

2

3

51L.3

4

5

6

7

1.

Title.

78-15671

8

Year:

9

0

1

2

3

4

5

from

For Amelia

The general approach

Following Aristotle,

editions.

view:

of this

On

the one hand, logic

book

to logic

remains the same as

in earlier

regard logic from two different points of an instrument or organon for appraising the

we is

correctness of reasoning; on the other hand, the principles and methods of logic used as organon are interesting and important topics to be themselves systematically investigated. This dual approach to logic

is

especially appro-

modern symbolic logic. Through the development of its special symbols, logic has become immeasurably more powerful an instrument for priate for

analysis

and deduction. And the principles and methods of symbolic logic are

through the study of logistic systems. Chapters 1 through 5 present the standard notations, methods, and principles of symbolic logic for use in determining the validity or invalidity of arguments. Successively more complex modes of argumentation are examined: compounds of simple first those whose validity turns on truth-functional

fruitfully investigated

statements, next those involving the simplest kinds of quantification, then more complex kinds of multiple quantification, and finally relational argu-

The standard methods of truth tables, rules of inference, conditional and indirect modes of proof, and quantification theory by way of ‘natural deduction’ techniques are introduced. The logic of relations is developed in a ments.

separate chapter that includes identity theory, definite descriptions, predicates of higher type, and quantification of predicate variables. Well over a

hundred new exercises have been added

in this edition to

help the student

acquire a practical mastery of the material. A transitional Chapter 6 explains the theory of deductive (or axiomatic)

which are the standard instruments for systematically investigating both mathematical and logical structures. To provide immediate contact with edition a very simple deductive system, what was Appendix B of the previous a new Chapter 7. Also, is brought forward to occupy the first two sections of important axiom to acquaint the student with a more complex, powerful, and systems,

7 presents the standard Zermelooffer a Fraenkel axiom system for set theory. Then the last three chapters chapters of systematic treatment of the logical principles used in the first five

system, the bulk of the

the book.

A

new Chapter

propositional calculus

is

developed according

to

the highest VII

Preface

viii

modern standards

of rigor,

and proved

be consistent and complete. Alter-

to

native notations and axiomatic foundations for propositional calculi are pre-

and then a The latter is shown

sented,

first

first-order function (or predicate) calculus

to

be equivalent

part of the book, and

is

also

There are three appendices: The

is

developed.

methods of the be consistent and complete.

to the ‘natural deduction’

proved

to

proves the deductive incompleteness

first

Chapter 3. This material occupied Section 3.4 of the previous edition, but it seemed to me (and to some others who used the book) that its presence there tended to interrupt the natural continuity desirable in a textbook. The second appendix develops Normal Forms and Boolean Expansions, which provide an algebraic treatment of truth-functional logic. The third appendix expounds the ramified of the 19 Ruies of Inference presented in the

first

two sections

of

theory of logical types as a method of resolving semantical paradoxes.

The

biggest change in this

set theory.

Of

new

edition

is

the presence of a

new chapter on

course, only an introduction to that vast subject matter can be

given in a single chapter. But

it

material for the following reasons: earlier editions suggested that

has seemed worthwhile to include that (1)

A number

some example

of people

who have

used

of an axiomatic system should be

included prior to the metalogically developed propositional and predicate calculi of the last three chapters.

I

think

it

an excellent suggestion and have

responded by including both the very simple axioms for Boolean Algebra in Section 7.2, and the more elaborate and powerful ZF axioms in subsequent sections of the new Chapter 7. (2) Some of the reasoning used to prove the deductive completeness of first-order logic in the final chapter makes use of set theoretic results, so

basis for the

it is

appropriate to present and explain the axiomatic

methods utilized

in the

book’s

last

chapter.

(3)

At

least part of the

importance of symbolic logic derives from its role in investigations into the foundations of mathematics. Set theory is another basic tool in this area: not an alternative but another equally essential instrument to be used in conjunction with symbolic logic in foundational studies. (4) Set theory is a discipline, clearly significant in its own right, in which the notations and methods of

symbolic logic are immediately,

and importantly applicable. So the presentation of set theory, in however limited a fashion, cannot but enhance the reader s appreciation of those notations and methods. (5) Set theory is an fruitfully,

area that the student of symbolic logic

is

uniquely prepared to enter immedi-

ately after mastering quantification theory, logicians

— especially

and more and more symbolic

those in mathematics departments

— are pursuing

their

researches in specialized and advanced parts of set theory. (6) Finally, set theory is intrinsically interesting and important in its own right, and I hope the brief introduction to

it

in this

new chapter

open the door for many paradise which Cantor has

will

students to enter into what Hilbert called ‘the created for us.’ 1

In addition to the

major change involving the addition of a new chapter,

Preface

ix

and those involved in relocating one appendix into that chapter and relegating the former Section 3.4 to the status of an appendix, there are other changes in this new edition that I hope will make the book more useful. These include some historical information about the development of symbolic logic, a more precise characterization of components and truth-functional components of compound statements, a clearer distinction between substitution (for variaand replacement (of equivalents), a simpler discussion of the rule of Indirect Proof and a clearer statement of its relationship with the strengthened principle of Conditional Proof, the addition of a needed restriction on the preliminary version of Universal Generalization, a new section on proving

bles)

whose propositions contain more than one quantifier, and later (in Chapter 5) an explanation of why that method cannot be used for relational arguments. Finally, with some help from friendly critics, a number of stylistic improvements have been made. As before, several teachers and students of logic have been kind enough to the invalidity of arguments

have given earnest consideration to all of the advice offered, even though 1 was not able to incorporate all of the changes proposed. For their helpful communications I am especially grateful to Professor Robert L. Armstrong, of The University of West Florida; Professor Eugene Atkin, of Central College, Iowa; Professor Walter Bass, of Indiana suggest ways to improve this book.

State University; Professor Robert

Marc

Professor

I

W. Beard,

of Florida State University;

Beaulieu, of College de Maisonneuve, Montreal, Quebec;

Anthony Belfiglio, of Chester State T. Blackwood, of Hamilton College, New

Professor Richard Beaulieu, of Paris;

College, Pennsylvania; Professor R.

York; Professor Alex Blum, of Bar-Ilan University, Israel; Professor Herbert C.

Bohnert, of Michigan State University; Professor Lorin Browning, of The College of Charleston, South Carolina; Professor Suehitra Dhar, of the Uni-

Northern Colorado; Professor James Edwards, of the University of Glasgow, Scotland; Professor Earl Eugene Eminhizer, of Youngstown State University, Ohio; Professor P. T. Geach, of The University of Leeds, England;

versity of

Professor James A. Gould, of

The University

Gary Leach, of Youngstown State

of South Florida;

Minogue, of University, Ohio; Terry L. Rankin, of Eastern Kentucky University; Professor Elizabeth Muir Ring, of Hamilton College, New York; Jennifer M. Shotwell, of Swarthmore College, Pennsylvania; Ronald C. Smith, of Asheville, North Mississauga, Ontario; Professor

Brendan

P.

Carolina; Professor Bangs L. Tapscott, of

The

University of Utah; Professor

Thomas, of Simpson College, Iowa; Professor Burke Townsend, of the University of Montana; and Professor Frank Williams, of Eastern Kentucky

Norman

L.

University.

Professor Paul Halmos, of Indiana University, and Professor Robert R. Stoll, of Oberlin College, read an early draft of the new Chapter 7 on set theory,

and made

a

number

and very helpful suggestions. have followed could, and any errors that remain are of course my

of criticisms

where I responsibility. The many omissions their advice

deliberate choice because the topics that

fill

I

1

chapter are a matter ot my own could not hope to cover in a single chapter all ol in the

hundreds of pages

in

the typical

book on the

subject.

x

Preface

This

new

edition has benefited also from

Alexander, Michael Bybee, and Dr.

Raymond

comments made by Dr. Linda Corsini.

Warm

thanks are due to

Kenneth J. Scott and Elaine W. Wetterau of the Macmillan Publishing Company for their expert editorial labor and unfailing helpfulness. Finally,

ing this

most of

new

all I

edition.

thank

my

wife for help and encouragement in prepar-

Introduction: Logic and

Language

1.1

What

1.2

The Nature

1.3

Truth and Validity 5 Symbolic Logic

1.4

Is

Logic? of

1

Argument

2

4

Arguments Containing Compound Statements Statements

Simple and

2.2

Conditional Statements

2.3

Argument Forms and Truth Tables 27 Statement Forms

2.4

3

Compound

2.1

The Method

of

19

32

Deduction

32

3.2 3.3

Proving Invalidity

3.4

3.5

The Rule The Rule

3.6

Proofs of Tautologies

3.7

The Strengthened Rule

3.8

Shorter Truth Table Technique

39

48 49

of Conditional Proof of Indirect Proof

52

54 of Conditional Proof

Absurdum Method

4

8

16

Formal Proof of Validity The Rule of Replacement

3.1

61

63

Singular Propositions and General Propositions

4.2

56

— Reductio ad

Quantification Theory 4.1

8

63

Proving Validity: Preliminary Quantification Rules

71

78

4.3

Proving Invalidity

4.4

Multiply-General Propositions

83 XI

xii

Contents

89

4.5

Quantification Rules

4.6

More on Proving

4.7

Logical Truths Involving Quantifiers

105

Invalidity

108

Logic of Relations 5.1

Symbolizing Relations

5.2

Arguments Involving Relations

5.3

Some

5.4

Identity and the Definite Description

5.5

Predicate Variables and Attributes of

130 134

Attributes of Relations

Attributes

6

116

140

150

Deductive Systems 6.1

Definition and Deduction

6.2

Euclidean Geometry

6.3 6.4

Formal Deductive Svstems 162 J Attributes of Formal Deductive Systems

6.5

Logistic Systems

157 158 164

166

Set Theory 7.1

7.2

170 175

7.3

Zermelo-F raenkel Set Theory (ZF)— The Axioms 176

7.4

Relations and Functions

7.5

Natural Numbers and the Axiom of Infinity Cardinal Numbers and the Choice Axiom

7.6 7.7

£3

The Algebra of Classes Axioms for Class Algebra

First Six

185

8.2

Object Language and Metalanguage Primitive Symbols and Well Formed

8.4

Formulas 215 Axioms and Demonstrations Independence of the Axioms

8.5

Development

8.6

Deductive Completeness

8.3

195

Ordinal Numbers and the Axioms of Replacement and Regularity 202

A Propositional Calculus 8.1

190

227 231

of the Calculus

250

237

213

Contents

9

Alternative

259

Systems and Notations 259 260

9.1

Alternative Systems of Logic

9.2 9.3

The Hilbert-Ackermann System 276 The Use of Dots as Brackets

9.4

A

9.5

The Stroke and Dagger Operators 282 The Nicod System

9.6

279

Parenthesis-Free Notation

281

290

A First-Order Function Calculus

no

10.2

290 The New Logistic System RSj 296 The Development of RS X

10.3

Duality

10.4

RS and the ‘Natural Deduction'

10.1

xiii

302

X

Techniques

307

10.5

Normal Forms

10.6

Completeness of RSj

10.7

RS with

10.8

First-Order Logic Including

X

Identity

311

318 328

ZF

Set

Theory

331

Appendix

A:

Incompleteness of the Nineteen Rules

333

Appendix

B:

Normal Forms and Boolean Expansions

337

Appendix

C:

The Ramified Theory

of

Types

344

Solutions to Selected Exercises

355

Special Symbols

381

Index

385

Introduction: Logic and Language What

1.1

Is

Logic?

According to 1 But Charles Peirce, ‘Nearly a hundred definitions of it have been given its Peirce goes on to write: ‘It will, however, generally be conceded that central problem is the classification of arguments, so that all those that are bad .... are thrown into one division, and those which are good into anothei It

is

easy to find answers to the question ‘What

is

Logic?

’.

The

study of logic, then,

is

the study of the methods and principles used in

distinguishing correct (good) from incorrect (bad) arguments. This definition is not intended to imply, of course, that only the student of logic can make the distinction. But the study of logic will help

one

to distinguish

between correct

and incorrect arguments, and it will do so in several ways. First of all, the proper study of logic will approach it as an art as well as a science, and the parts of the theory being learned. Here, as anywhere else, practice will help to make perfect. In the second place, the exact study of logic, especially symbolic logic, like the study of any other study oi science, will tend to increase proficiency in reasoning. And finally, the

student will do exercises in

all

validity ol all logic will give students certain techniques for testing the when arguments, including their own. This knowledge is of value because

mistakes are easily detected, they are less likely to be made. Logic has frequently been defined as the science of reasoning. That definiaccurate. tion, although it gives a clue to the nature of logic, is not quite

Reasoning are

is

that special kind of thinking called inferring, in

drawn from premisses. As

which conclusions

thinking, however, reasoning

is

not the special

Psyprovince of logic, but part of the psychologist’s subject matter as well. complex chologists who examine the reasoning process find it to be extremely that are and highly emotional, consisting of awkward trial and error procedures

insight.

These are

all

—

—and

sometimes apparently irrelevant flashes ol not of importance to psychology. Logicians, however, are

illuminated by sudden

correctness of interested in the actual process of reasoning, but rather with the

Logic, in Dictionary of Philosophy Publishing Co., Inc., New York, 1925. 1

and Psychology James Mark Baldwin,

ed.,

Macmillan

,

1

2

Introduction: Logic and

[Ch.

Language

the completed reasoning process. Their question

is

1

always: Does the conclusion

reached follow from the premisses used or assumed? If the premisses provide adequate grounds for accepting the conclusion, if asserting the premisses to be true warrants asserting the conclusion to be true also, then the that

is

reasoning

is

correct. Otherwise, the reasoning

and techniques have been developed primarily

methods the purpose of making this

incorrect. Logicians

is

for

distinction clear. Logicians are interested in all reasoning, regardless of

its

subject matter, but only from this special point of view.

1.2

The Nature

Inferring

of

Argument

an activity

is

which one proposition

in

is

affirmed on the basis of

one or more other propositions that are accepted as the starting point of the process. The logician is not concerned with the process of inference, but with the propositions that are the initial and end points of that process, relationships

and the

between them.

Propositions are either true or false, and in this they differ from questions,

commands, and exclamations. Grammarians classify the linguistic formulations of propositions, questions, commands, and exclamations as declarative, interrogative, imperative, and exclamatory sentences, respectively. These are familiar notions. It is important to distinguish between declarative sentences and the propositions they may be uttered to assert. A declarative sentence is always part of a language, the language in which it is spoken or written, whereas propositions are not peculiar to any of the languages in which they may be expressed. Another difference between declarative sentences and propositions

is

that the

same sentence may be uttered

assert different propositions. (For

uttered by different persons to

in different contexts to

example, the sentence T

make

am hungry’ may

be

different assertions.)

The same sort of distinction can be drawn between sentences and statements. The same statement can be made using different words, and the same sentence can be uttered in different contexts to make different statements. The terms proposition and ‘statement are not exact synonyms, but in the writings of logicians they are used in much the same sense. In this book, both terms will be used. In the following chapters, cially in

and

Chapters 2 and

5) to refer to

we

shall also use the

and the term ‘proposition (especially in Chapters 4 the sentences in which statements and propositions are 3)

expressed. In each case the context should

Corresponding

to

make

every possible inference

these arguments that logic

is

chiefly concerned.

is

clear

which are alleged

what

is

meant.

an argument, and

it

is

with

An argument may be defined as

any group of propositions or statements, of which one the others,

term ‘statement (espe-

is

claimed

to follow

from

provide grounds for the truth of that one. In ordinary usage, the word ‘argument' also has other meanings, but in logic it has to

the technical sense explained. In the following chapters

argument

also in a derivative sense to refer to

we

shall use the

word

any sentence or collection of

The Nature

Sec. 1.2]

sentences in which an argument

is

formulated or expressed.

of

Argument

When we

do,

3

we

be presupposing that the context is sufficiently clear to ensure that unique statements are made or unique propositions are asserted by the utterance of shall

those sentences.

Every argument has a structure, in the analysis of which the terms ‘premiss’ and ‘conclusion’ are usually employed. The conclusion of an argument is that proposition which is affirmed on the basis of the other propositions of the argument. These other propositions, which are affirmed as providing grounds or reasons for accepting the conclusion, are the premisses of that argument. note that ‘premiss’ and ‘conclusion’ are relative terms, in the sense that

We

the same proposition can be a premiss in one argument and a conclusion in another. Thus the proposition All humans are mortal is premiss in the argu-

ment All

humans

Socrates

is

are mortal.

human.

Therefore, Socrates

and conclusion

in the

is

mortal.

argument

All animals are mortal.

humans

All

Therefore,

Any

are animals. all

humans

are mortal.

proposition can be either a premiss or a conclusion, depending upon

its

assumed

for

context.

It is

a premiss

when

it

occurs in an argument in which

it is

some other proposition. And it is a conclusion when it an argument that is claimed to prove it on the basis of other

the sake of proving

occurs

in

propositions that are assumed.

between deductive and inductive arguments. All arguments involve the claim that their premisses provide some grounds for the truth of their conclusions, but only a deductive argument involves the claim It is

customary

to distinguish

premisses provide absolutely conclusive grounds. The technical terms character‘valid’ and ‘invalid’ are used in place of ‘correct and incorrect in premisses izing deductive arguments. A deductive argument is valid when its

that

its

and conclusion are so related that it is absolutely impossible for the premisses logic is to to be true unless the conclusion is true also. The task of deductive and conclarify the nature of the relationship that holds between premisses clusion in a valid argument,

and

to

provide techniques for discriminating valid

from invalid arguments. provide Inductive arguments involve the claim only that their premisses opposite some grounds for their conclusions. Neither the term valid nor its arguments difler ‘invalid’ is properly applied to inductive arguments. Inductive their premin the degree of likelihood or probability that

among isses

themselves

in indueconfer upon their conclusions. Inductive arguments are studied

Introduction: Logic and

4

Language

[Ch.

we

1

be concerned only with deductive use the word ‘argument’ to refer to deductive arguments however,

tive logic. In this book,

arguments, and shall

shall

exclusively.

Truth and Validity

1.3

Truth and falsehood characterize propositions or statements and said to characterize the declarative sentences in

may also be

which they are formulated.

Arguments, however, are not properly characterized as being either true or false

but as valid or invalid

2 .

There

is

a connection

between the

validity or

argument and the truth or falsehood of its premisses and conclusion, but the connection is by no means a simple one. Some valid arguments contain true propositions only, as, for example, invalidity of an

All bats are All

mammals.

mammals have

Therefore,

bats have lungs.

all

An argument may

lungs.

contain false propositions exclusively and

still

be

valid, as,

for example,

All trout are

All

mammals.

mammals have

Therefore,

all

wings.

trout

have wings.

argument is valid because if its premisses were true, its conclusion would have to be true also, even though, in fact, they are all false. These two examples show that although some valid arguments have true conclusions, not all of them This

do.

The

validity of an

argument does

not, therefore, guarantee the truth of

its

conclusion.

When we

consider the argument

am President, then I am I am not President. Therefore, am not famous. If

I

famous.

I

we can is

see that although both premisses

invalid. Its invalidity

and conclusion are

becomes obvious when

it

is

true, the

argument

compared with another

argument of the same form:

“Some

logicians use the

adopt that usage ‘invalid

to

in

(

term

‘valid to characterize

We

statements that are logically true. shall chapter 10, Section 10.6. Until then, however, we applv the terms ‘valid’ and

arguments exclusively.

Symbolic Logic

Sec. 1.4] If

Rockefeller

Rockefeller

is

is

President, then Rockefeller

sion

is false.

ments have

is

famous.

not President.

Therefore, Rockefeller

This argument

is

is

not famous.

clearly invalid because

The two

latter

false conclusions,

premisses are true but

its

its

conclu-

examples show that although some invalid argunot

all

of

them

do.

The falsehood of its conclusion

does not guarantee the invalidity of an argument. Rut the falsehood of conclusion does guarantee that either the argument its

premisses

is

its

invalid or at least one of

is false.

An argument must sion. It

must be

termed

‘sound’.

valid,

satisfy

and

all

two conditions of

its

to establish the truth of

its

conclu-

premisses must be true. Such an argument

To determine the truth

scientific inquiry in general, since at all.

5

or falsehood of premisses

premisses

may

is

is

the task of

deal with any subject matter

But determining the validity or invalidity

of

arguments

is

the special

province of deductive logic. The logician is interested in the question of validity even for arguments that might be unsound because their premisses

might happen to be false. A question might be raised about the legitimacy of that suggested that logicians should confine their attention to true premisses only.

It is

might be arguments that have interest. It

often necessary, however, to depend

upon

the validity

whose premisses are not known to be true. Scientists test their theories by deducing from them conclusions that predict the behavior of observable phenomena in the laboratory or observatory. The conclusion is then tested directly by observation of experimental data and if it is true, the results confirm the theory from which the conclusion was deduced. If the conclusion is of arguments

the results disconfirm or refute the theory. In either case, the scientist is testable convitally interested in the validity of the argument by which the false,

deduced from the theory being investigated, for if that argument is oversimplification invalid, his whole procedure is without point. Although an impoi taut of scientific method, our example shows that questions of validity ai e even for arguments whose premisses are not true.

clusion

1.4

is

Symbolic Logic

that has been explained that logic is concerned with arguments and arguments contain propositions or statements as their premisses and concluThese premisses and conclusions are not linguistic entities, such as It

sions.

what declarative sentences aie typically however, uttered to assert. The communication of propositions and arguments, Arguments requires the use of language, and this complicates our problem. declarative sentences, but are, rather,

to English or any other natural language are often difficult in which they appraise because of the vague and equivocal nature of the words

formulated

in

6

Language

Introduction: Logic and

[Ch.

1

are expressed, the ambiguity of their construction, the misleading idioms they

may

and

contain,

their pleasing but deceptive

of the

argument

laries.

The

resolu-

still

is

remains.

To avoid the peripheral workers

style.

not the central problem for the logician, however, for they are resolved, the problem of deciding the validity or invalidity

tion of these difficulties

even when

metaphorical

difficulties

connected with ordinary language,

have developed specialized technical vocabueconomizes the space and time required for writing his

in the various sciences

The

scientist

reports and theories by adopting special symbols to express ideas that

would

otherwise require a long sequence of familiar words to formulate. This has the

amount of attention needed, for when a equation grows too long, its meaning is more difficult to grasp. The

further advantage of reducing the

sentence or

introduction of the exponent symbol in mathematics permits the expression of the equation

AxAxAxAxAxAxAxAxAxAxAxA =BXBXBXBXBXBXB more

briefly

and

intelligibly as

A 12 = B 7 A

advantage has been obtained by the use of graphic formulas in organic chemistry. The language of every advanced science has been enriched by like

similar symbolic innovations.

A

special technical notation has

made

been developed

use of certain abbreviations to facilitate his

symbolic logic augmented symbols.

this

for logic as well. Aristotle

own

Modern many more special

investigations.

base by the introduction of

he difference between the old and the new logic is one of degree rather than of kind, but the difference in degree is tremendous. Modern symbolic logic has become immeasurably more powerful a tool for analysis and 1

deduction through the development of its own technical language. The special symbols of modern logic permit us to exhibit with greater clarity the logical structures of arguments that may be obscured by formulation in ordinary language.

arguments into the valid and the invalid when they are expressed in a special symbolic language, for with symbols the peripheral problems of vagueness, ambiguity, idiom, metaphor, and amphiboly do not arise. The introduction and use of special symbols serve not only to It is

easier to divide

facilitate the appraisal of

arguments, but also to clarify the nature of deductive

inference. I

he logician

s

special symbols are

much

better adapted to the actual

drawing of inferences than is ordinary language. Their superiority in this respect is comparable to that of Arabic over the older Roman numerals for purposes

compute

of

computation.

the product of

It is

easy to multiply 148 by 47, but very difficult to

CXLVIII and XLVII.

Similarly, the

drawing of

Symbolic Logic

Sec. 1.4]

inferences and the evaluation of arguments

is

7

greatly facilitated by the adop-

To quote Alfred North Whitehead, an the advance of symbolic logic:

tion of a special logical notation.

important contributor to

.

.

.

bv the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye, which otherwise would call into play the 3 higher faculties of the brain.

symbolic logic systematically rather than historically, a few historical remarks may be appropriate at this point. Since the 1840s, symbolic logic has developed along two different historical paths. One

Although

this

book

treats

them began with the English mathematician George Boole (1815-1864). Boole applied algebraic notations and methods first to symbolize and then to validate arguments of the kind studied by Aristotle in the fourth century b.c.

of

This route

may be

characterized as an effort to apply mathematical notations

and methods to traditional, nonmathematical kinds of arguments. The other path began with the independent efforts of the English mathematician Augustus De Morgan (1806-1871) and the American scientist and philosopher Charles Peirce (1839-1914) to devise a very precise notation for relational arguments. The earlier logic had largely ignored this type of argument, which, nevertheless, plays a central role in mathematics. This historical path

characterized as an effort to create a

new quasi-mathematical kind

may be

of logical

notation and analytical technique for use in mathematical derivations and

demonstrations.

works of the German mathematician and philosopher Gottlob Frege (1848-1925), the Italian mathematician Guiseppe Peano (1858-1932), and the English philosophers Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970), whose These two

Principle logic.

historical paths coalesced in the brilliant

Mathematica was an important landmark

Some

in the history of

of Boole’s contributions are reported in the

first

two

symbolic

sections of

The contributions of the others have become so thoroughly incorporated into modern symbolic logic that only occasional references to their more distinctive ideas are appropriate.

Chapter 7 and

3

in

A. N. Whitehead,

1911.

Appendix

An

B.

Introduction

to

Mathematics, Oxford University Press, Oxford, England,

Arguments Containing Compound Statements Simple and Compound Statements

2.1

be divided into two kinds, simple and compound. A one that does not contain any other statement as a compo-

All statements can

simple statement

is

compound statement does contain another statement component part. For example, ‘Atmospheric testing of nuclear weapons be discontinued or this planet will become uninhabitable' is a compound

nent part, whereas every as a

will

statement that contains, as

components, the two simple statements ‘Atmos-

its

pheric testing of nuclear weapons will be discontinued and

become

The component themselves be compound, of course. uninhabitable’.

As the term ‘component a larger statement

is

a

is

parts of a

‘this

planet will

compound statement may

used in logic, not every statement that

component

of the larger statement.

is

part of

For example, the

last

words of the statement, ‘The third wife of Bertrand Russell was a beautiful girl can be regarded as a statement in its own right. But it is not a component part or a component of the larger statement containing it. For a part of a statement to be a component of a larger statement, two conditions must be satisfied. First, the part must be a statement in its own right; and second, if the six

,

part

is

replaced

in the larger

statement by any other statement, the result of

that replacement

must be meaningful. Although the first condition is satisfied in the example given, the second is not. For if the part Bertrand Russell was a beautiful girl

replaced by the statement ‘Where there’s smoke, there’s the result of that replacement is arrant nonsense 1 is

fire’,

.

The statement Roses are red and violets are blue is a conjunction a compound statement formed by inserting the word ‘and between two statements. Iwo statements so combined are called conjuncts. The word ‘and has ,

other uses, however, as in the statement ‘Castor and Pollux were twins’, which is not compound, but a simple statement asserting a relationship. We introduce the dot

•

as a special

symbol

for

combining statements conjunctively. Using

account of compound and Component statements has been suggested by Professor of Northern Illinois University, Professor Alex Blum of Bar-Ilan University, and Professor James A. Martin of the University of Wyoming. See Martin’s ‘How Not to Define I

C.

Bis

Mason Myers

Truth-Functionality,’ Logique et Analyse, 1970, 14 e Annee, N° 52, pp. 476-482. 8

Simple and Compound Statements

Sec. 2.1]

this notation, the

blue’.

preceding conjunction

Where p and q

are any

is

9

written ‘Roses are red* violets are

two statements whatever,

their conjunction

is

written p*q.

Everv statement

is

either true or false, so

we

can speak of the truth value of

where the truth value of a true statement is true and the truth value of a false statement is false. There are two broad categories into which compound statements can be divided, according to whether or not there is anything other than the truth values of its component statements that determines the truth value of the compound statement. The truth value of the conjunction of two statements is completely determined by the truth value of

a statement,

conjuncts.

its

A

conjunction

is

true

if

both

otherwise. For this reason a conjunction

statement, and

its

its is

conjuncts are true, but false

a truth-functional

conjuncts are truth-functional components of

compound it.

For example, the truth-value of the compound statement ‘Percival Lowell believed that Mars is inhabited’ is not in any way determined by the truth value of its component simple statement ‘Mars is inhabited’. This is so because it could be true that

Not every compound statement

Percival Lowell believed that Mars

is

is

truth-functional.

inhabited, regardless of whether

it

is

person can believe one truth without believing another, and a person can believe one falsehood without believing all of them. So the component ‘Mars is inhabited is not a truth-functional component of the inhabited or not.

A

Percival Lowell believed that

compound statement

Mars

is

inhabited,

and

compound statement. We define an occurrence of a component of a compound statement to be a truth-functional component of that compound statement provided that: It that occurrence of the component is replaced in the compound (or in any component of the compound statement which contains the component in the latter statement

is

not a truth-functional

question) by different statements that have the same truth value as each other,

the different

compound

statements produced by these replacements will also

have the same truth values as each other. And now a compound statement is defined to be a truth-functional compound .statement, if all of its components are truth-functional components of the statement.

The only compound statements we shall consider here will be truth-functionally compound statements. Therefore, in the rest of this book we shall use the term “simple statement” to refer to any statement that tionally

not truth-func-

compound.

Since conjunctions are truth-functionally

symbol

is

is

compound

statements, our dot

Given any two statements p and q of truth values they can have, and in every case

a truth-functional connective.

there are just four possible sets

the truth value of their conjunction p-q

is

uniquely determined. The four

possible cases can be exhibited as follows:

\\ hat Is with the one proposed by Professor David H. Sanford in his 14' Annee, N° 52, pp. 483-486. a Truth Functional Component?’ Logique et Analyse, 1970, 2

Compare

this definition

10

Arguments Containing Compound Statements in case

is

in

is

p case p

in case

p

is

in case

p

is

and (j true and (j false and (j false and q true

is is is is

[Ch. 2

p'q is true; false, p'q is false; true, p*q is false; false, p'q is false. true,

Representing the truth values true and

bv the capital

false

letters

T

and

‘F’,

which the truth value of a conjunction is determined by the truth values of its conjuncts can be displayed more briefly by means of

way

respectively, the

in

a truth table as follows:

Since

it

P