An Introduction to Formal Logic

Reviewed by Corey Maley, Assistant Professor, University of Kansas on 8/21/16

Comprehensiveness rating: 5

This book is a comprehensive introduction to formal logic. Although it does not have an index, the table of contents is sufficient to provide the reader with an idea of where to find various topics. This book would be useful for a one-semester course in introductory logic, and should allow students to become comfortable with metatheory in later classes.

Content Accuracy rating: 5

I found no errors in the textbook, although there were some points where some might disagree--or at least have questions--about the author's descriptions and exercises. For example, when asked to translate "Of course the Duchess is lying!" using D to stand for "The Duchess is lying", one might wonder whether the original expression is translatable. Is "Of course" truth-functional? Perhaps, but there is room for discussion. That may be the author's intent, but it is not clear. Overall, however, the content is free of errors.

Relevance/Longevity rating: 5

Logic is, almost by definition, timeless, so this book will be useful for some time. Any updates should be easy to incorporate.

Clarity rating: 3

There are some places where I found the books clarity somewhat lacking, particularly for the novice student.

One example is when the author discusses metatheory. I found the discussion potentially confusing for some students. The notions of an object language and a metalanguage are familiar enough to philosophers, but not necessarily to beginning logic students. On pages 29 and 30, the discussion is quite compressed, and the wording might be confusing to some. For example, the author states that the metalanguage will be "mathematical English," but what that refers to is not made clear. The author then uses bold, stylized A and B for metavariables, which I will write in this review as @ and %, given that I cannot reproduce the font here. So the author states the following:

"It is important here that @ is not the sentence letter A. Rather, it is a variable that stands in for any wff at all. Notice that this variable @ is not a symbol of SL, so ¬@ is not an expression of SL."

Then later:

"For instance, if @ and % are wffs of SL, then (@ & %) is a wff of SL."

This is a subtle discussion in general, and difficult to explain well in any textbook. I fear that this discussion would be confusing to some students.

There are other small, but potentially problematic, areas where the book could be more clear. For example, the author switches from using T and F to stand for "true" and "false" in the second chapter, to 1 and 0 afterwards. The explanation given is that these are just arbitrary symbols, so it doesn't matter what one uses. That indicates to me that 1 and 0 should have been used from the beginning.

Another example is where the author discusses truth-functional connectives on p. 38. Rather than list some clear examples of truth-functional connectives, the author immediately discusses examples of connectives that are not truth functional, and then mentions the diamond operator in modal logic (a topic that is not discussed in any detail within the book). This is an unnecessary tangent.

A third example is on p. 49. The first example of quantified predicate logic the author discusses is one that ends up being translatable and valid in sentential logic, so predicate logic ends up being unnecessary. That seems like a bad choice for the very first example, since this is not usually the case when one is using predicate logic. Furthermore, the author does not clearly discuss why this particular example is translatable using only sentential logic.

The final example I will mention is found on p. 51. The discussion of definite descriptions is interesting, but seems a bit out of place. The author notes that there is much philosophical discussion of issues regarding singular terms, proper names, and definite descriptions. But it seems to me that this is presenting a lot of information that is potentially confusing before we have yet encountered some simple examples.

Consistency rating: 4

The book is largely consistent, except for the change from using T and F to 1 and 0.

Modularity rating: 5

The book is as modular as a text in introductory logic can be. I would imagine that students who have some familiarity with sentential logic, for example, would have no trouble going straight into the later sections. However, the very nature of this kind of material makes complete modularity nearly impossible.

Organization/Structure/Flow rating: 4

The books overall structure is quite good. I have only two comments. First, some people might prefer proofs to come in a slightly different order. For example, some might prefer that, after introducing sentential logic, proofs in sentential logic are covered. Then, after predicate logic, proofs in predicate logic are covered. The author chooses to present proofs in one chapter. There is nothing wrong with this choice, but it may be easier for some students to have proofs broken up into more than one chapter.

Second, as mentioned above, there are some points where the "flow" of the book is interrupted by what I take to be unnecessary tangents, or at the very least, discussions that should come later in the text.

Interface rating: 5

The interface of this book has no problems whatsoever.

Grammatical Errors rating: 5

I found no grammatical errors.

Cultural Relevance rating: 5

The book is culturally sensitive. My only comment here is that some students may not be familiar with what "a standard deck of cards" refers to (which occurs in one of the translation problems). However, knowledge of that is not necessary to complete the problem (it might just seem very odd).

The problems and exercises in this book are very good, and go beyond what is normally found in introductory logic books. I think that some of these problems would be especially useful for students who are interested in going on to more advanced logic courses. For example, rather than just translations and proofs, the author includes questions that ask students to think about logic (implicitly, at least) at a metatheoretic level.