Introduction to Formal Logic PDF
Preview Introduction to Formal Logic
Introduction 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 OXFORD UNIVERSITY PRESS Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and certain other countries. Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America. © 2018 by Oxford University Press For titles covered by Section 112 of the US Higher Education Opportunity Act, please visit www.oup.com/us/he for the latest information about pricing and alternate formats. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by license, or under terms agreed with the appropriate reproduction rights organization. Inquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above. You must not circulate this work in any other form and you must impose this same condition on any acquirer. 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.
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