simple formal logic

358 Pages·2010·3.27 MB·English

SIMPLE FORMAL LOGIC
with Common-Sense Symbolic Techniques
Arnold vander Nat
Loyola University Chicago
First published 2010
by Routledge
270 Madison Avenue, New York, NY 10016
Simultaneously published in the UK
by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
Routledge is an imprint of the Taylor & Francis Group, an informa business
This edition published in the Taylor & Francis e-Library, 2009.
To purchase your own copy of this or any of Taylor & Francis or Routledge’s
collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.
© 2010 Taylor & Francis
All rights reserved. No part of this book may be reprinted or reproduced
or utilized in any form or by any electronic, mechanical, or other means,
now known or hereafter invented, including photocopying and recording,
or in any information storage or retrieval system, without permission in
writing from the publishers.
Trademark Notice: Product or corporate names may be trademarks or
registered trademarks, and are used only for identification and explanation
without intent to infringe.
Library of Congress Cataloging in Publication Data
Vander Nat, Arnold.
Simple formal logic : with commn-sense symbolic techniques / Arnold
vander Nat.
p. cm.
Includes index.
1. Logic. I. Title.
BC71.V36 2009
160–dc22 2009001506
ISBN 0-203-87452-8 Master e-book ISBN
ISBN10: 0–415–99745–3 (hbk)
ISBN13: 978–0–415–99745–4 (hbk)
CONTENTS
Preface ix
CHAPTER 1 BASIC LOGICAL CONCEPTS 1
1.1 Introduction 1
1.2 Arguments 8
1.3 Evaluating Arguments 21
1.4 Classification of Sentences 27
1.5 Proofs, Inconclusive and Erroneous Arguments 36
1.6 Deductive and Inductive Validity 44
CHAPTER 2 PROPOSITIONAL LOGIC 52
2.1 Introduction to Propositional Logic 52
2.2 Details about Negations 61
2.3 Details about Conditional Sentences 69
Reference Sheet for Conditional Sentences 76
2.4 Determining the Truth-Value of Sentences 80
2.5 Truth-Tables to Test the Validity of Arguments 85
2.6 Truth-Table Tests for Other Logical Concerns 93
2.7 Presentation of the Rules of Deduction 103
Reference Sheet for the Propositional Rules of Deduction 113
2.8 Making Deductions for Arguments 116
2.9 Strategy in Deductions 125
Twenty Sample Deductions 131
More Advanced Topics in Propositional Logic 135
2.10 The Rules of Conditional Proof and Indirect Proof 135
2.11 Further Uses of Deductions 141
vi Contents
CHAPTER 3 TRADITIONAL LOGIC 148
3.1 Introduction: New Perspectives 148
3.2 Categorical Sentences in Traditional Logic 150
3.3 English Variations on Categorical Forms 156
3.4 Venn Diagrams for Categorical Sentences 171
3.5 Venn Diagrams for Syllogisms and Equivalences 177
3.6 Equivalence Operations 185
3.7 Deductions in Traditional Logic 193
Reference Sheet for the Rules of Traditional Logic 206
3.8 Combining Propositional Logic with Syllogistic Logic 211
CHAPTER 4 MODERN QUANTIFICATIONAL LOGIC 216
4.1 Introduction to Quantificational Logic 216
4.2 Types of English Simple Sentences and Other Details 225
Symbolizing 100 Quantificational Sentences 231
4.3 The Truth-Value of Quantificational Sentences 236
4.4 Deductions in Quantificational Logic 240
Reference Sheet for the Rules of Quantificational Logic 255
More Advanced Topics in Quantificational Logic 259
4.5 Deductions with Conditional Proof and Indirect Proof 259
4.6 Demonstrating Invalidity 263
4.7 Relational Sentences 266
4.8 Deductions with Relational Sentences 273
4.9 Working with Identities 279
CHAPTER 5 LOGICAL FALLACIES 286
5.1 Introduction 286
5.2 Fallacies of Irrelevance 290
1. Argumentum ad Baculum 290
2. Argumentum ad Hominem 291
3. Argumentum ad Populum 292
4. Argumentum ad Verecundiam 293
5. Argumentum ad Misericordiam 295
6. Argumentum ad Ignorantiam 296
7. Tu Quoque Fallacy 297
8. Guilt by Association Fallacy 298
9. Red Herring Fallacy 299
Contents vii
5.3 Fallacies of Misconstrual 301
10. Straw Man Fallacy 301
11. Post Hoc Ergo Propter Hoc Fallacy 302
12. Fallacy of Special Pleading 303
13. Fallacy of Improper Generalization 304
14. Fallacy of Improper Instance 305
15. Complex Question Fallacy 306
16. Fallacy of Syntactic Ambiguity 308
17. Fallacy of Semantic Equivocation 309
18. Fallacy of Division 311
19. Fallacy of Composition 312
5.4 Fallacies of Presumption 314
20. False Dichotomy Fallacy 314
21. Slippery Slope Fallacy 315
22. Fallacy of Ad Hoc Reasoning 316
23. Petitio Principii Fallacy 318
24. Fallacy of Inconsistent Premisses 319
Glossary and Index of Terms 325
List of Symbols and Abbreviations 345

PREFACE
This logic book is designed for a first course in logic, and as the title suggests, a course in
formal logic. Teachers often find themselves in a quandary about what logic text to use for
their introductory course. They want to teach a real course in logic, that treats the laws and
the methods of logic, to ordinary students across the curriculum, but they do not want to
teach a more advanced course in formal logic. This is precisely the objective that this logic
book is designed to achieve.
The purpose of this logic course is to give students a comprehensive knowledge of the laws of
logic, together with the method of logical deduction that uses these laws, in such a manner that
people can make use of this knowledge in their ordinary reasoning. Of course, many other
interesting and useful topics are covered as well (as the Table of Contents shows), but the
focus is on the laws and method of logic. This focus, by itself, makes this course a course
in formal logic (in contrast to what is normally called informal logic), but naturally, it is
formal logic in another sense as well. Logic is presented as a precise symbolic system, divided
into the three areas of Propositional Logic, Traditional Logic, and Quantificational Logic.
The nature of this symbolic system, is, of course, the very issue that creates the educational
quandary. Our purpose requires that the symbolic system be formulated to correspond to
the patterns and methods of ordinary reasoning. This may sound like a truism that hardly
needs to be stated, but that is not so. Many introductory logic texts present logic systems
that use special rules and formal techniques that do not correspond to ordinary reasoning,
such as the now commonly presented method of semantic (truth) trees with open or closed
branches to determine logical validity, which is an elegant and efficient technique, and an
important tool for solving system-related theoretical questions, but which is artificial and
unnatural, and ironically, incapable of being used in ordinary reasoning. By contrast, our
presentation of logic gives students exact formal tools that are readily used in ordinary
reasoning. The symbolic techniques will be natural, and the rules of logic will be ample,
familiar, and readily usable.
This book, then, presents logic as a formal, symbolic system, and includes a comprehensive
presentation of the laws of logic and the method of logical deduction, all such as
x Preface
correspond to patterns of ordinary reasoning. The three areas of Propositional Logic, Trad-
itional Logic, and Quantificational Logic are presented, and treated in a standard way,
except for Traditional Logic. Traditional Logic is standardly treated with much archaic detail
and as an unserviceable truncated system, that has some rules but no method of deduction.
Our treatment of Traditional Logic trims off all the archaic detail and replaces it with a
simple, comprehensive, and commonsense method of syllogistic deduction identical to the
deductive methods of the other areas of logic. This new treatment integrates Traditional
Logic into the deductive framework of modern logic as a partner on an equal footing with
the other areas, which is something that many teachers of logic will find to be a very
satisfying result.
The book has been designed to make good use of many exercises spaced throughout the
book. All of the sections of the chapters include various sets of exercises, typically of varying
degrees of difficulty. To achieve the pedagogical goals of the book, the students should
be required to do many of these exercises on a daily basis. There are specially designed
worksheets on the Routledge Companion Website (see information below) for these exercises
that facilitate the completion, as well as the correction, of these exercises. It is also recom-
mended that the daily lectures review the exercise material to some extent.
There is more material in this book than what is needed to have a good introductory course
in logic. The material has been arranged so that the instructor can selectively choose various
parts to achieve the desired content, the desired level of difficulty, and the desired pace of
the course. The book has the following main divisions:
Chapter 1. Basic logical concepts
Chapter 2. Propositional Logic
Chapter 2. More advanced topics in Propositional Logic
Chapter 3. Traditional Logic
Chapter 4. Modern Quantificational Logic
Chapter 4. More advanced topics in Quantificational Logic
Chapter 5. Logical Fallacies
The order of the chapters is flexible. In particular, the chapters on Propositional Logic
and Traditional Logic are independent of each other, and one may therefore reverse their
order if desired. As a further alternative, since the chapters on Propositional Logic and
Quantificational Logic are complete treatments of logic, one may simply opt not to do the
chapter on Traditional Logic. But our treatment of that material is a much improved and
useful version, and from the viewpoint of application, this chapter is highly recommended.
The chapter on Logical Fallacies is a stand-alone chapter, and one may take it at any point
after Chapter 1, or not at all. There is, then, significant flexibility in how one may use this
book. Here are some examples:
Preface xi
(1) For a more moderate course, do all the chapters 1 through 5, but skip the advanced
sections of both chapters 2 and 4.
(2) For a more relaxed course, do the material suggested for (1), except, also skip section 2.6
(additional uses of truth tables), section 3.8 (combined deductions of Propositional and
Traditional Logic), and section 4.4 (quantificational deductions).
(3) For a more intense course, do the entire book, chapters 1 through 4, including the
advanced sections, but with Chapter 5 optional.
(4) For a course in formal logic, do chapters 1, 2, and 4, including the advanced sections,
with Chapter 3 optional, and skipping Chapter 5.
(5) For a course in applied logic, do chapters 1, 2, 3, and 5, with the advanced sections of
Chapter 2 optional, and skipping Chapter 4.
Enjoy the book.
Arnold vander Nat
Routledge Companion Website
The Routledge companion website for this book contains important supplemental
materials to aid students in their study of logic. It is located at:
http://www.routledge.com/textbooks/9780415997454
Its key features are:
1. The website contains all the Exercise Worksheets for each of the exercises in the book.
Throughout the book, the student is asked to use these Exercise Worksheets, which are a
great convenience for both the students and the teacher. These worksheets are printable
PDF files, and the student may download them one at a time, or in entire chapter
batches.
2. In addition, this website contains a special Deduction Strategy Module that helps stu-
dents learn effective strategy techniques in building deductions for arguments in both
Propositional and Traditional Logic. This module also provides students with practice in
learning the rules of deduction. The program tabulates a score for each problem as well
as a score for the whole session.
3. The website also contains printable study supplements, such as flashcards for symboliza-
tions, flashcards for the deduction rules, and summary rule sheets. All of these are
downloadable PDF files.
4. Finally, this website contains the Instructor’s Manual for this book. Verified teachers with
authorized passwords will be able to access the Instructor’s Manual from the website.