Table Of ContentIntroduction to Formal Logic
Introduction to Formal Logic
RUSSELL MARCUS
Hamilton College
The question of logic is: Does the conclusion
certainly follow if the premises be true?
AUGUSTUS DE MORGAN
Formal Logic: Or, The Calculus of Inference, Necessary and Probable (1847)
New York Oxford
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Library of Congress Cataloging-in-Publication Data
Names: Marcus, Russell, 1966– author.
Title: Introduction to formal logic / Russell Marcus.
Description: New York : Oxford University Press, 2018. | Includes
b ibliographical references.
Identifiers: LCCN 2017051737 (print) | LCCN 2017053175 (ebook) | ISBN
9 780190861797 (Ebook) | ISBN 9780190861780 (pbk.)
Subjects: LCSH: Logic—Textbooks.
Classification: LCC BC108 (ebook) | LCC BC108 .M35 2017 (print) | DDC
160—dc23
LC record available at https://lccn.loc.gov/2017051737
9 8 7 6 5 4 3 2 1
Printed by LSC Communications, United States of America
Contents
Preface vii
Chapter 1: Introducing Logic 1
1.1: Defining ‘Logic’ 1
1.2: Logic and Languages 3
1.3: A Short History of Logic 5
1.4: Separating Premises from Conclusions 9
1.5: Validity and Soundness 16
Key Terms 21
Chapter 2: Propositional Logic: Syntax and Semantics 22
2.1: Logical Operators and Translation 22
2.2: Syntax of PL: Wffs and Main Operators 43
2.3: Semantics of PL: Truth Functions 48
2.4: Truth Tables 59
2.5: Classifying Propositions 68
2.6: Valid and Invalid Arguments 77
2.7: Indirect Truth Tables 83
2.8: Notes on Translation with PL 105
Key Terms 112
Chapter 3: Inference in Propositional Logic 113
3.1: Rules of Inference 1 113
3.2: Rules of Inference 2 124
3.3: Rules of Equivalence 1 135
3.4: Rules of Equivalence 2 146 v
vi CONTENTS
3.5: Practice with Derivations 156
3.6: The Biconditional 164
3.7: Conditional Proof 174
3.8: Logical Truths 184
3.9: Indirect Proof 191
3.10: Chapter Review 203
Key Terms 211
Chapter 4: Monadic Predicate Logic 212
4.1: Introducing Predicate Logic 212
4.2: Translation Using M 219
4.3: Syntax for M 233
4.4: Derivations in M 238
4.5: Quantifier Exchange 254
4.6: Conditional and Indirect Proof in M 263
4.7: Semantics for M 273
4.8: Invalidity in M 280
4.9: Notes on Translation with M 299
Key Terms 309
Chapter 5: Full First-Order Logic 310
5.1: Translation Using Relational Predicates 310
5.2: Syntax, Semantics, and Invalidity in F 328
5.3: Derivations in F 337
5.4: The Identity Predicate: Translation 351
5.5: The Identity Predicate: Derivations 370
5.6: Translation with Functions 380
5.7: Derivations with Functions 390
Key Terms 400
Appendix A: Fallacies and Argumentation 401
Appendix B: The Logical Equivalence of the Rules of Equivalence 413
Summary of Rules and Terms 419
Solutions to Selected Exercises 421
Glossary/index 475
Preface
Introduction to Formal Logic (IFL) and Introduction to Formal Logic with Philosophi-
cal Applications (IFLPA) are a pair of new logic textbooks, designed for students of
formal logic and their instructors, to be rigorous, yet friendly and accessible. Unlike
many other logic books, IFL and IFLPA both focus on deductive logic. They cover
syntax, semantics, and natural deduction for propositional and predicate logics. They
emphasize translation and derivations, with an eye to semantics throughout. Both
books contains over 2000 exercises, enough for in-class work and homework, with
plenty left over for extra practice, and more available on the Oxford website.
WHY THIS LOGIC BOOK?
I initially conceived my project as a two-part logic book. The first part would be a
thorough, standard introduction to formal logic: syntax, semantics, and proof theory
for propositional and predicate logics. The second part would add interesting exten-
sions of the basic formal material and engaging reflections on why philosophers are
interested in logic, with essay prompts and suggestions for further readings. These
two parts reflect how I teach logic, asking students both to work through the formal
material and to write a little about how logic is useful outside of logic.
As the book that I initially envisioned went through the review process at Oxford,
it became clear that some instructors were mainly interested in the first part, and did
not see a use for the second. The book you are holding, Introduction to Formal Logic,
is one result: a nuts-and-bolts introductory formal deductive logic textbook. There
is a brief introductory chapter. Chapter 2 covers propositional semantics, leading to
the standard truth-table definition of validity. Chapter 3 covers natural deductions
in propositional logic. Chapter 4 covers monadic predicate logic. Chapter 5 covers
full first-order logic. This material is straight logic, and I have kept the text simple and
focused, without distracting discussions of the philosophy of logic. (The other book,
IFLPA, contains the same formal material, but adds enrichment sections that encour-
age reflecting on the technical work and integrating writing into logic classes. See
below for more details on the differences between the two books.)
vii
viii PREFACE
Teachers of logic are often faced with a bimodal distribution of student abili-
ties: some students get the material quickly, and some students take more time—
sometimes significantly more time—to master it. Thus one central challenge to logic
teachers is to figure out how to support the former group of students while keeping
the latter group engaged. I have addressed this challenge, in part, by providing lots
of exercises with varying, progressive levels of difficulty and including some exercise
sections that can be used by the strongest students and skipped by others without
undermining their later work.
Since logic is most often taught in philosophy departments, special attention is
given to how logic is useful for philosophers. Many examples use philosophical con-
cepts, translating philosophical arguments to one of the formal languages, for ex-
ample, and deriving their conclusions using the inferential tools of the text. Some of
these arguments are artificial, as one might expect in an introductory logic text; I do
not endorse their content. I hope mainly to have the arguments be ones that someone
might use. There are plenty of exercises with more ordinary content, too, which may
be friendlier to the beginning student, or one with no background in philosophy.
SPECIAL FEATURES
Each section of IFL contains a Summary and a section of important points to
Keep in Mind.
Key terms are boldfaced in the text and defined in the margins, and are listed
at the end of each chapter. In addition, all terms are defined in a glossary at
the end of the book.
There are over 2000 exercises in the book.
Exercises are presented progressively, from easier to more challenging.
Translate-and-derive exercises are available in every section on deriva-
tions, helping to maintain students’ translation skills.
Translation exercises are supplemented with examples for translation
from formal languages into English.
Regimentations and translations contain both ordinary and philosophi-
cal themes.
Solutions to exercises, about 20% of total, are included at the back of the
book. Solutions to translate-and-derive exercises appear in two parts:
first, just the translation, and then the derivation. Solutions to all exer-
cises are available for instructors.
IFL contains several topics and exercise types not appearing in many standard
logic textbooks:
Seven rules for biconditionals, parallel to the standard rules for conditionals;
Exercises asking students to interpret and model short theories
Two sections on functions at the end of chapter 5
PREFACE ix
Exercises asking students to determine whether an argument is valid or
invalid, or whether a proposition is a logical truth or not, and then to
construct either a derivation or a counterexample
These sections are perfect for stronger students, while easily skipped by
others.
Emphasis is placed on semantics through the text, with truth tables for prop-
ositional logic and interpretations and models for first-order logic.
Two supplementary sections on subtleties of translation, 2.8 and 4.9, pro-
vide students with discussions of the complications of translation while not
interfering with the progress of the formal work.
An appendix on fallacies and argumentation supports instructors’ connec-
tions between formal logic and real-world reasoning.
INTRODUCTION TO FORMAL LOGIC OR INTRODUCTION TO
FORMAL LOGIC WITH PHILOSOPHICAL APPLICATIONS? TWO
BOOKS—YOUR CHOICE.
In addition to this formal logic book, I have written a longer version: Introduction to
Formal Logic with Philosophical Applications (IFLPA) with two chapters not included
in IFL. These additional chapters contain thirteen enrichment essays with writing
prompts for students, and reading suggestions. The topics of these sections include
conditionals, modal logic, three-valued logics, deduction and induction, logic and sci-
ence, logic and philosophy of religion, logic and the philosophy of mind, truth, names
and definite descriptions, and others. These sections are independent of the formal
logic in chapters 1–5, and of each other.
All enrichment essays in IFLPA encourage students to reflect on the philosophi-
cal applications of their work in formal logic. I use the material in class as biweekly
pauses in formal instruction, which I call Philosophy Fridays. I ask students to write
an essay each term in addition to their homework and exams. My approach has helped
to engage students and their individual interests, and to manage more effectively the
natural diversity of skills in a typical logic class. With more enrichment material avail-
able than I ordinarily use in a semester, I vary my choices in each class, sometimes
responding to student interest.
I have included in IFL three sections of IFLPA, tucked away at the ends of chapters:
2.8 and 4.9 on some interesting subtleties of translation, and an appendix on falla-
cies and argumentation. This enrichment material need not get in a logic instructor’s
way. But if your students begin to reflect on what they are doing and ask questions
about why our logic is as it is, or why philosophers are interested in it, you might find
IFLPA to be of some use. The formal material is the same in IFL and IFLPA: the same
examples, the same exercises, and the same numberings. So instructors and students
may work together with either version and move freely between the two books.
