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Logic Made Easy: How to Know When Language Deceives You

Deborah J. Bennett
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[Reviewed by
Robert S. Wolf
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Logic Made Easy: How to Know When Language Deceives You, by Deborah Bennett, is a solid, nicely written, and well organized attempt to explain and teach the basic principles of logic to a lay audience. It is a short book, with only about 200 pages of actual text, and I would call it a rather "easy read," especially considering the subject matter. Indeed, one of the most impressive aspects of this book is that it manages to tread a fine line quite effectively: it avoids excessive technicalities and detail, and at the same time avoids the temptation to be more witty and entertaining than informative. The cover jacket claims that the book "explains the timeless concepts of logic in a contemporary and user-friendly voice," and I believe that this claim is generally accurate.

The book has a clear and deliberate structure that is largely based on syntax, at least for a while. Chapter 2 is about universally quantified statements, while Chapter 3 discusses negations. Chapter 4 brings in existential statements, Chapter 6 introduces conditionals, and Chapter 7 talks about conjunctions and disjunctions. There are thirteen chapters in all, and the later chapters are not organized around particular connectives or quantifiers.

One of the foremost questions in my mind when I encounter such a book is to ask who will be reading it. What is the intended audience? Who would benefit most from it? The very first paragraph of the introduction provides some sort of answer, albeit confusing, to these questions: "Logic Made Easy is a book for anyone who believes that logic is rare. It is a book for those who think they are logical and wonder why others aren't. It is a book for anyone who is curious about why logical thinking doesn't come 'naturally.' It is a book for anyone who wants to be more logical." Only the last sentence of this description mentions the obvious audience of people who would like to improve their logic skills. It sounds as if this book is written mainly for logical people who can't understand why everyone else isn't as logical as they are. I believe that this introduction is misleading in that the emphasis is backwards: while the book cites many studies that give some insight into why logical reasoning is difficult for many, the primary focus of the book is to improve the reader's ability to reason logically.

Perhaps one way to clarify the book's likely audience more precisely is to mention some things the book is not. For one thing, it is definitely not a textbook. It lacks the structure of a text, including exercises. On the other hand, as I have already mentioned, it is not a book whose primary purpose is to entertain the reader, although it is certainly entertaining in spots. It is also not a mathematics book, or a book about mathematical logic. Indeed, the author deliberately avoids assuming any mathematical sophistication on the part of the reader. For instance, she calls Euclid's proof that there is an infinite number of primes "too involved" to include.

Based on these considerations, I can see three primary audiences for this book. One would be general, "lay" readers who simply want to improve their logical skills in a pleasant way. The second would be scientists and engineers, and also students in these areas (high school or undergraduate), who have the same goal. This second group could also include students in many disciplines who want to improve their "critical thinking" skills, as well as lower division undergraduate mathematics students. The third group would be educators and social scientists — psychologists, sociologists, etc. — who have a professional interest in understanding the obstacles to logical thinking and how to help others overcome those obstacles.

Here are a few examples of this book's helpful explanations of logical concepts. Confusion between a statement and its converse is probably the most common error made in basic logical reasoning. On page 43, Bennett clarifies this point nicely, using "vice versa" as a sort of synonym for converse: "You might think of the converse as the vice versa. All faculty members are employees of the university, but not vice versa. All dogs love their owners and vice versa. (Although I'm not sure either is true.)" A purist might object to the inclusion of quantifiers in these statements, but I think that's excusable a book of this type. Later, on page 109, she makes the interesting point that confusion over converses can lead to errors in understanding the meaning of conditional probabilities. Specifically, the difference between "the probability of A, given B" and "the probability of B, given A" can get muddled by someone who does not fully understand the logic of conditionals. This could be a serious problem for doctors reading reports on the efficacy of diagnostic tests, for example.

Bennett provides many examples of the differences between the strictly logical meanings of words and their meanings in ordinary speech. In the chapter on the existential quantifier, for instance, she points out that the word "some" often means "some but not all" in common parlance, but it never has this meaning to a logician (page 64). Later in the same chapter, (page 67), she mentions that the statements "Some dogs are poodles" and "Some poodles are dogs" have the identical logical content, but they would have very different connotations in ordinary speech. In the chapter on conditionals, she provides a thorough discussion of the many meanings that the words "if ... then" can have in practice, in contrast to their precise and sometimes unintuitive meaning in logic.

Bennett also presents a variety of logical puzzles and tests to educate and entertain the reader. The well-known "Wason selection task" involving four cards, devised by the cognitive psychologist Peter Wason in the 1960s, is described in the chapter on conditionals. Another of Wason's tests is given on page 120: you are shown a black diamond, a white diamond, a black circle, and a white circle. You are told that the black diamond is a THOG. You are also told that a THOG is by definition an object that has a particular (unknown) color OR a particular (unknown) shape, BUT not both. For example, a THOG might be defined as an object that is white or a diamond, but not both. Given this information, what conclusions can be drawn about whether the other three objects are or are not THOG's? This test demonstrates the subtlety involved with the exclusive "or." Many subjects give wrong answers about all three objects!

The THOG puzzle could be of interest to educators and social scientists, as could many of the tests and experiments described in this book. One of the topics that is visited several times in the book is the elusive relationship between logic and subject matter. Theoretically, there should be no relationship: "Logical reasoning is supposed to take place without regard to either the sense or the truth of the statement or the material being reasoned about" (page 49). But very few people can make an absolute separation, and in practical situations it is usually better not to separate them. Pure logic is less powerful in daily life than a combination of logic, context, experience, etc. — what we call "common sense." Under the heading "Familiarity — Help or Hindrance?", the book gives examples in which extra-logical considerations help people draw the correct logical conclusion, but it also gives examples in which such considerations interfere with the correct use of logic. Bennett returns to this theme later in the book (page 181), citing a study that showed that "faulty logic" may result less from the inability to use logic correctly than from various types of interference. This interference can arise from outside knowledge or preconceived notions or even the attitudes or emotional involvement of the reasoner. If this is true, then helping people to eliminate these sources of interference could be one of the most efficient ways to help them be more logical.

All in all, this is a well written, interesting, and entertaining book. People from many backgrounds, including those with an interest in mathematics, should be able to enjoy it and learn from it.

Robert S. Wolf is professor of mathematics at the California Polytechnic State University in San Luis Obispo.

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