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A Concise Introduction to Logic A Concise Introduction to Logic A Concise Introduction to Logic Craig DeLancey Open SUNY Textbooks © 2017 Craig DeLancey ISBN: 978-1-942341-42-0 ebook 978-1-942341-43-7 print This publication was made possible by a SUNY Innovative Instruction Technology Grant (IITG). IITG is a competitive grants program open to SUNY faculty and support staff across all disciplines. IITG encourages development of innovations that meet the Power of SUNY’s transformative vision. Published by Open SUNY Textbooks Milne Library State University of New York at Geneseo Geneseo, NY 14454 This book was produced using Pressbooks.com, and PDF rendering was done by PrinceXML. A Concise Introduction to Logic by Craig DeLancey is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted. Dedication For my mother Contents About the Textbook Reviewer's Notes Adam Kovach 0. Introduction 1 Part I: Propositional Logic 1. Developing a Precise Language 7 2. “If…then….” and “It is not the case that….” 15 3. Good Arguments 29 4. Proofs 45 5. “And” 55 6. Conditional Derivations 69 7. “Or” 79 8. Reductio ad Absurdum 89 9. “… if and only if …”, Using Theorems 99 10. Summary of Propositional Logic 113 Part II: First Order Logic 11. Names and predicates 119 12. “All” and “some” 127 13. Reasoning with quantifiers 139 14. Universal derivation 147 15. Relations, functions, identity, and multiple quantifiers 159 16. Summary of first order logic 173 Part III: A Look Forward 17. Some advanced topics in logic 181 Bibliography 209 About the Author 210 About Open SUNY Textbooks 211 About the Textbook A Concise Introduction to Logic is an introduction to formal logic suitable for undergraduates taking a general education course in logic or critical thinking, and is accessible and useful to any interested in gaining a basic understanding of logic. This text takes the unique approach of teaching logic through intellectual history; the author uses examples from important and celebrated arguments in philosophy to illustrate logical principles. The text also includes a basic introduction to findings of advanced logic. As indicators of where the student could go next with logic, the book closes with an overview of advanced topics, such as the axiomatic method, set theory, Peano arithmetic, and modal logic. Throughout, the text uses brief, concise chapters that readers will find easy to read and to review. Reviewer's Notes Adam Kovach True to its name, A Concise Introduction to Logic, by Craig DeLancey, surveys propositional logic and predicate logic and goes on to introduce selected advanced topics, in little over 200 pages. The book provides an integrated presentation of basic syntactic and semantic concepts and methods of logic. Part I starts with the concept of a formal language. The concept of valid inference, truth tables and proofs are introduced immediately after the first two propositional connectives. Connectives and inference rules are introduced in alternation, to develop a complete simple natural deduction system for propositional logic. Part II, adds the apparatus of quantification and proof rules for a complete predicate logic. The text covers the logic of relations, sentences with multiple quantifiers and Russell’s theory of definite descriptions. The presentation of concepts and principles is orderly, clear and thought provoking. Many topics are introduced with examples of philosophical arguments drawn from classic sources, adding depth of knowledge to an introductory course. The first two parts end with systematic overviews. The focus is on formal deductive logic throughout. Informal fallacies and traditional syllogistic logic are not covered. Advanced topics covered in the final part of the text include an axiomatic approach to logic, mathematical induction, a deduction theorem for propositional logic, and brief introductions to set theory, modal logic and number theory. The reviewer, Adam Kovach, is Associate Professor of Philosophy at Marymount University in Arlington, VA, where he teaches courses in many subjects including logic. 0. Introduction 0.1 Why study logic? Logic is one of the most important topics you will ever study. “How could you say such a thing?” you might well protest. And yet, consider: logic teaches us many things, and one of these is how to recognize good and bad arguments. Not just arguments about logic—any argument. Nearly every undertaking in life will ultimately require that you evaluate an argument, perhaps several. You are confronted with a question: Should I buy this car or that car? Should I go to this college or that college? Did that scientific experiment show what the scientist claims it did? Should I vote for the candidate who promises to lower taxes, or for the one who says she might raise them? And so on. Our lives are a long parade of choices. When we try to answer such questions, in order to make the best choices, we often have only one tool: an argument. We listen to the reasons for and against various options, and must choose between them. And so, the ability to evaluate arguments is an ability that is very useful in everything that you will do—in your work, your personal life, your deepest reflections. If you are a student, note that nearly every discipline, be it a science, one of the humanities, or a study like business, relies upon arguments. Evaluating arguments is the most fundamental skill common to math, physics, psychology, literary studies, and any other intellectual endeavor. Logic alone tells you how to evaluate the arguments of any discipline. The alternative to developing these logical skills is to be always at the mercy of bad reasoning and, as a result, you will make bad choices. Worse, you will always be manipulated by deceivers. Speaking in Canandaigua, New York, on August 3, 1857, the escaped slave and abolitionist leader Frederick Douglass observed that: Power concedes nothing without a demand. It never did and it never will. Find out just what any people will quietly submit to and you have found out the exact measure of injustice and wrong which will be imposed upon them, and these will continue till they are resisted with either words or blows, or with both. The limits of tyrants are prescribed by the endurance of those whom they oppress.[1] We can add to Frederick Douglass’s words that: find out just how much a person can be deceived, and that is just how far she will be deceived. The limits of tyrants are also prescribed by the reasoning abilities of those they aim to oppress. And what logic teaches you is how to demand and recognize good reasoning, and so how to avoid deceit. You are only as free as your powers of reasoning enable. 0.2 What is logic? Some philosophers have argued that one cannot define “logic”. Instead, one can only show logic, by doing it and teaching others how to do it. I am inclined to agree. But it is easy to describe the benefits of logic. For example, in this book, you will learn how to: 1 • Identify when an argument is good, and when it is bad; • Construct good arguments; • Evaluate reasons, and know when they should, and should not, be convincing; • Describe things with a precision that avoids misunderstanding; • Get a sense of how one can construct the foundations of arithmetic; • Begin to describe the meaning of “possibility” and “necessity”. That is by no means a complete list of the many useful things that logic can provide. Some of us believe that logic and mathematics are ultimately the same thing, two endeavors with the same underlying structure distinguished only by different starting assumptions. On such a view, we can also think of logic as the study of the ultimate foundations of mathematics. (This is a reasonable characterization of logic, but those afraid of mathematics need not fear: logic must become quite advanced before its relation to mathematics becomes evident.) Ultimately, the only way to reveal the beauty and utility of logic is to get busy and do some logic. In this book, we will approach the study of logic by building several precise logical languages and seeing how we can best reason with these. The first of these languages is called “the propositional logic”. 0.3 A note to students Logic is a skill. The only way to get good at understanding logic and at using logic is to practice. It is easy to watch someone explain a principle of logic, and easier yet to watch someone do a proof. But you must understand a principle well enough to be able to apply it to new cases, and you must be able to do new proofs on your own. Practice alone enables this. The good news is that logic is easy. The very goal of logic is to take baby steps, small and simple and obvious, and after we do this for a long while we find ourselves in a surprising and unexpected new place. Each step on the way will be easy to take. Logic is a long distance walk, not a sprint. Study each small step we take, be sure you know how to apply the related skills, practice them, and then move on. Anyone who follows this advice can master logic. 0.4 A note to instructors This book incorporates a number of features that come from many years of experience teaching both introductory and advanced logic. First, the book moves directly to symbolic logic. I don’t believe that informal logic is worth the effort that it requires. Informal logic largely consists of memorization (memorizing seemingly disconnected rules, memorizing fallacies, and so on). Not only is this sure to be the kind of thing that students will promptly forget, but it completely obscures the simple beauty of why the various rules work, and why the fallacies are examples of bad reasoning.
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