# 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:

- 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. A student who learns symbolic logic, however, is learning a skill. Skills are retained longer; they encourage higher forms of reasoning; and they have far more power than a memorized list of facts. Once one can recognize what makes an argument good, one can recognize the fallacies, regardless of whether one has memorized their names.

Second, this book focuses on some of the deeper features of logic, right at the beginning. The notions of semantics and syntax are introduced in the first chapter. Ideas like theorem, and a model, are discussed early on. My experience has shown that students can grasp these concepts, and they ultimately pay off well by greatly expanding their own understanding.

Third, this book uses examples, and constructs problems, from our intellectual history in order to illustrate key principles of logic. The author is a philosopher, and understands logic to be both the method of philosophy and also one of the four fundamental sub-disciplines of philosophy. But more importantly, these examples can do two things. They make it clear that arguments matter. Weighty concerns are discussed in these arguments, and whether we accept their conclusions will have significant effects on our society. Seeing this helps one to see the importance of logic. These examples can also make this book suitable for a logic course that aims to fulfill a requirement for an introduction to the history of thought, an overview of Western civilization, or the knowledge foundations of a related discipline.

Fourth, I follow a no-shortcuts principle. Most logic textbooks introduce a host of shortcuts. They drop outer parentheses, they teach methods for shrinking truth tables, and so on. These moves often confuse students, and for no good reason: they have no conceptual value. I suspect they only exist to spare the impatience of instructors, who would like to write expressions and truth tables more quickly. In this book, except in the last chapter that looks to advanced logic, we will not introduce exceptions to our syntax, nor will we spend time on abridged methods. The only exception is writing “T” for true and “F” for false in truth tables.

Fifth, this book includes a final chapter introducing some advanced topics in logic. The purpose of this chapter is to provide students with some understanding of the exciting things that they can study if they continue with logic. In my experience, students imagine that advanced logic will be just more proofs in first order logic. Giving them a taste of what can come next is valuable. My hope is that this chapter will motivate students to want to study more logic, and also that it can serve as a bridge between their studies in basic logic and the study of advanced logic.

Finally, about typesetting: quotation is an important logical principle, and so I adopted the precise but comparatively rare practice of putting punctuation outside of quotes. This way, what appears in the quotations is alone what is being defined or otherwise mentioned. I use italics only to indicate the meaning of a concept, or to distinguish symbolic terms of the object language from functions of the object language. Bold is used to set aside elements of our metalanguage or object language.

## 0.5 Contact

The author would appreciate any comments, advice, or discoveries of errata. He can be contacted at: craig.delancey@oswego.edu

## 0.6 Acknowledgements

The typesetting of proofs used the lplfitch LaTex package developed by John Etchemendy, Dave Barker-Plummer, and Richard Zach.

Thanks to two reviewers for the Open SUNY Textbook program; and to Allison Brown and the other people who help make the Open SUNY Textbook program work; and to Carol Kunzer, Karen Gelles, and Carrie Fishner for copy editing the text. Thanks to Derek Bullard for catching some errata.

[1] From Blassingame (1985: 204), in a speech titled “The Significance of Emancipation in the West Indies.”