# An Introduction to Formal Logic

P.D. Magnus, University of Albany, State University of New York

Pub Date: 2012

ISBN 13:

Publisher: Fecundity

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This textbook is a comprehensive overview of sentential logic and first order quantified logic. It begins by introducing basic notions such as the … read more

This textbook is a comprehensive overview of sentential logic and first order quantified logic. It begins by introducing basic notions such as the nature of arguments and deductive validity. It continues by introducing the language of sentential logic and then truth tables. It proceeds with a discussion of the language of quantified logic, and then follows this with a discussion of formal semantics. The book concludes by introducing a system for natural deduction. Along the way it provides a number of practice problems with solutions to selected problems found in the back of the book. It also contains a helpful appendix giving an overview of logical symbols.

I found the book to be accurate and the explanations to be clear and helpful.

As this is a more or less standard sort of textbook on propositional logic and predicate logic, there is not much danger that its content will become outdated. Where textbooks like this sometimes seem dated is in the kinds of English sentences they use for translation exercises. But this textbook avoids using such potentially datable examples and updates to it could be made easily.

The book is remarkably clear and accessible to individuals with no previous background in logic. In fact, I would rate it as giving the most lucid explanations of the concepts it covers that I have found to date. I was especially impressed with how clearly it managed to explain basic formal semantics.

I have no concerns about the book’s consistency. Basic terminology and notation is introduced and then consistently adhered to throughout.

For the most part, the book is about as modular as a logic textbook can be (the nature of the subject matter does constrain the order of presentation to some extent). It is clearly divided into chapters and sections with a helpful table of contents. One could use the textbook even if one wanted to present things in a different order than did the author. For example, though the author introduces formal semantics before discussing natural deduction, it would not be too difficult to use this textbook while reversing that order.

The book is organized well and the concepts it introduces build on one another. In some places, the author makes different choices about the order of presentation than I would make (e.g. the author introduces formal semantics before natural deduction). But such choices have to be made and other instructors may prefer how the author made them.

I found no difficulties with the book’s interface. The chapters and section headings are well laid out and the typeset is clear.

I find no issues with the book’s grammar.

As far as I am able to discern, the textbook is not culturally biased. The examples it contains are generic enough to be accessible to people of a variety of backgrounds and the book is free of racist, sexist, and classist language.

Frankly, I think this is an excellent logic textbook. I highly recommend it.

This concise text accomplishes what it sets out to achieve. It manages to cover the material of sentential logic up through quantificational logic … read more

This concise text accomplishes what it sets out to achieve. It manages to cover the material of sentential logic up through quantificational logic right up to the point of setting up the problem of completeness. It does not include independent treatments of categorical logic, informal fallacies, or inductive logic, let alone other topics, but it is not designed to treat those topics. As it stands, it is suitable for a one-semester course on sentential and quantificational logic.

The book is highly accurate and precise, particularly in the author’s discussions of translating from English into quantificational logic. Whereas many textbooks more or less confine themselves to providing specimens of translations and then counsel the student to “get the feel” for it, this textbook goes into detailed discussions justifying the standard choices. As such, ‘Chapter 4’ alone would serve admirably as a supplement to another textbook or as a replacement for the topic of translations. Be warned, however, that the author elects to the use the term “sentence” to designate what many textbooks designate with the term “statement.” This choice also affects the value that the book might otherwise have as part of a general educational program that would complement a course in logic with another in general grammar and rhetoric. Some other details might be mentioned. For example, on page 6, the author offers exceedingly brief lists of premise and conclusion indicators. The instructor might feel the need to amplify these lists one way or the other. Further, on page 6, the author states that the conclusion comes at the end of the series of sentences that compose it. While this is obviously true for any argument that has been reworked so as to conform to the standard form, it is obviously not true of many arguments that appear in ordinary language contexts. On page 9, the author introduces the concept of a “good” argument without mentioning that of sound or cogent arguments. Minimally, one would expect some mention of soundness at least in this connection, as the book is a text on deductive logic.

The book contains material highly relevant to the study of sentential and quantificational logic. The brief treatments of models and natural deduction are very helpful in how the author interrelates the two topics. His approach of introducing n-tuple predicate relations prior to quantifiers is refreshing and, I think, potentially helpful to the student. One cannot envision the text becoming quickly outmoded or obsolete.

The author’s prose is as clear as it could be. Indeed, his writing is so clear and concise at points that one can anticipate undergraduates new to the subject to struggle with learning directly from the text, as no, or very few, concessions are made to the kind of muddy thinking that often characterizes a person’s first study of this subject. As is true of many logic textbooks, one must already be a rather clear thinker in order to learn from it. However, it falls on the teacher to make the material more accessible during class discussion, albeit accessible to that degree that inherently difficult material permits.

The books is wholly internally consistent in terms of terminology and its framework.

The text is broken down into six chapters and thirty-four sections. It is highly adaptable to a variety of pedagogical plans. Each chapter, however, does build on the preceding.

The topics of this text follow an excellent and, in my experience, a partially original design. The author moves back and forth between sentential and quantificational logic when this suits his presentation and the overall movement of the book is towards a comprehensiveness that does not leave any threads hanging.

There are no problems with images, charts, symbolism, etc. All is in fact remarkably clear.

There seems to be an error on the top of page 116, where the word 'define' seems out of place. There is one typo on page 80, where 'specific' presumably should be 'specify' and another of page 72 where 'if' should be 'is.' Other than these instances, the grammar is impeccable.

The text is of universal relevance as it treats of some of the basic forms of reasoning. As such, it is relevant in any cultural context.

The book will be particularly useful for philosophy majors who need a textbook geared towards equipping them with a knowledge of some of the formal machinery of quantificational logic. It is worth noting that 'Chapter 6' presents natural deduction in systemic fashion, laying out introduction and elimination rules relative to each of the basic operators before presenting derivative rules (modus ponens, hyp. syll., etc.). The text does not bother with some oft used forms, such as constructive dilemma and exportation.

The book covers most of the topics needed for an introduction to logic class. In sections 6.9 a glimpse into metalogic is offered. The book does not … read more

The book covers most of the topics needed for an introduction to logic class. In sections 6.9 a glimpse into metalogic is offered. The book does not suggest nevertheless any connection to informal logic (other textbooks in logic abound in this area). It would be useful to relate logic to critical thinking

The book is accurate. Some typos exist, but it is clear that the author is willing to correct them.

The book reflects the state of the art in sentential and predicate logic with natural deduction. A chapter or a section could mention other methods such as tableaux and offer an alternative.

The writing of the author is exceptionally clear. Sometimes it may look dry and formal, but the book is clear and the language is not ambiguous.

As a textbook in logic this textbook excels in consistency. Notation is consistent, language is consistent. The author makes some attempts to show alternative notations.

The text is modular but the reader finds fewer references to other textbooks or to the very rich history of logic.

Overall it is good. But some students may find problems in transitioning from sentential to predicate logic. The book does not attempt to show the extreme importance of predicate (QL) logic for science, philosophy etc. The book should explain why logic is important outside ordinary language. And examples for other areas than everyday language would be beneficial

The text flows fine, but there are no hyperlinks from chapter to chapters. Navigation is sometimes awkward.

Nothing to add: excellent grammar and structure of sentences.

Logic and critical thinking are both national deficits, to paraphrase a bumper sticker. This text should be present in any college -- or i am wrong-- probably in every high school in USA.

Overall it is good. But some students may find problems in transitioning from sentential to predicate logic. The book does not attempt to show the extreme importance of predicate (QL) logic for science, philosophy etc. The book should explain why logic is important outside ordinary language. And examples for other areas than everyday language would be beneficial

This textbook is very good at covering the basics one would expect to find an an introductory logic course that focuses on deductive logic. It lays … read more

This textbook is very good at covering the basics one would expect to find an an introductory logic course that focuses on deductive logic. It lays things out very clearly and offers concise explanations that I think students would appreciate. It is a traditional formal logic text and would serve as well as any of the well-known logic texts that are similarly aimed. This book does what it does in a way that students would find straightforward. As it stands, I usually end up pulling from several texts and sources (both open source and traditional) for teaching a logic course. This volume would not get rid of the need for multiple resources for how I teach intro logic. There are topics that are not covered here that I end up teaching in a course on logic, particularly about inductive logic, scientific reasoning, and Mill's Methods, that are not present in this volume. There are also approaches to logic that students find engaging - like courtroom examples and logical fallacies - that are not covered. The author is pretty frank on this point: "We will not be interested in inductive arguments in this book" (page 10). I'm rating the book at 5 on comprehensiveness because it fulfills its goals, but it's important to note that teachers who include inductive reasoning as part of their logic courses might only use this volume as one resource and not the only class text.

I did not observe any errors, but I did not work every problem as I looked through this book.

Introductory formal logic shouldn't change rapidly, and this book covers many of the basic topics in deductive logic. It should have a very long life.

The book is clear and introduces many concepts in a succinct manner. Students will appreciate the author's approach.

Terms are consistent, and the structure really works.

One of the best things about this book is that someone could "remix" the order of the book and not confuse students.

The organization of this volume is easy to follow. One of the other reviewers lamented that there wasn't an index, but an index isn't needed for a PDF where one can use the "find" feature to find whatever keyword you hope to search for. This is actually one feature that makes the PDF version stronger than a printed version for students hoping to find what they want easily.

I was surprised at how convenient the PDF was to manage. I think it's great that the textbook is concise, which lends itself to easier flipping and searching. I thought I would hate it, but actually found it more convenient than a traditional text.

I did not find any errors in grammar.

I did not find any culturally insensitive text in this textbook.

I would highly recommend inclusion of this text as a replacement for many formal logic books. Its brevity would be appreciated by students.

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 … read more

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.

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.

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

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.

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

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.

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.

The interface of this book has no problems whatsoever.

I found no grammatical errors.

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.

The text covers propositional logic (symbolization, truth tables and proofs) and predicate logic (symbolization, semantics, and proofs). There is a … read more

The text covers propositional logic (symbolization, truth tables and proofs) and predicate logic (symbolization, semantics, and proofs). There is a short appendix on alternate symbolizations (including Polish notation), and another which gives answers to selected exercises. There is also a "Quick Reference" section giving definitions of the basic sentence operators, symbolic expressions for standard natural language forms, and all the rules of inference. It doesn't contain any "extras" (material on definition, fallacies, etc.)but all the basics of formal logic are there.

The book contains few errors.

The content of elementary formal logic does not change.

The text is as clear as many others. In my experience students generally need the course instructor to "interpret" text material in logic.

The text is internally consistent.

The text is divided into 6 chapters: basic logical concepts; symbolization in propositional logic; truth table; symbolization in predicate logic; semantic theory for predicate logic; proofs. Chapter 6 on proofs first presents proofs in propositional logic, so it would be possible to proceed from truth tables (chapter 3) directly to proofs in propositional logic (secs. 1-3 of chapter 6).

The organization of the text is fine.

I found no interface issues.

The text contains no grammatical errors.

I did not find any racist or sexist examples, or any others offensive to me.

Overall, this is a very satisfactory text--especially considering the cost of commercially published texts. One thing to be aware of, however, is that the instructor generally will need to supplement the homework exercises. There are not enough exercises, and they tend not to be presented in graduated levels of difficulty. This is especially true in the chapter on proofs. This chapter (on proofs) is also a very terse presentation and could use more development with examples and discussion. On the other hand, there are some very good thought exercises, particularly in the earlier chapters.

Though concise, the book is comprehensive: it covers all the topics one would normally discuss in an introductory logic course, with both sentential … read more

Though concise, the book is comprehensive: it covers all the topics one would normally discuss in an introductory logic course, with both sentential and quantificational logic--syntax and semantics, truth tables, natural deduction. The book has no index, but the table of contents should suffice. Key terms are defined at the end of each chapter.

No errors. The proof system is, in fact, both sound and complete, for example.

This is elementary logic; the basics will not change. Obsolescence is not an issue.

The book is very clearly written, and admirable for its concision. Technical terms tend to be introduced less formally at first, with rigorous necessary-and-sufficient conditions provided later; this is a nice way to ease the student in.

The book maintains the same notational conventions consistently throughout (and those conventions are helpfully summarized in an appendix).

The content could easily be shuffled around to suit individual instructors' preferences. For instance, the book covers semantics (informally) for sentential and quantificational logics before covering those languages' syntax; but the syntactic sections are clearly separated, and so could be presented first, if that were the instructors' preference. In addition, natural deduction is covered last, in Chapter 6, after a full chapter on formal semantics. Instructors could easily reverse the order.

The book is well-organized. Different people may prefer to introduce topics in a different order, but that can be accommodated.

No issues.

No errors that I saw.

No issues here. References to Batman and Lemmy (RIP), e.g., don't seem problematic to me.

The book's main attractions are its lucidity and brevity. Almost all logic books, in my experience, are too long; they encourage the illusion that anyone can pick up the relevant understanding and skills autodidactically. But only a very small group of high-aptitude people can do that; the rest need the help of an instructor. That this book is brief, then, is an advantage: students who (ill-advisedly) re-read many times at least won't waste too much time doing so. (Also, what they're reading is admirably clear, so that helps.) But the brevity does place a burden on the instructor--to supplement, sometimes heavily, succinct explanations in the text. For example, a discussion of "Proof Strategy" in section 6.6 takes up less than one and a half pages--and that's meant to cover natural deduction for both sentential and quantificational logic. Any experienced instructor knows that much more discussion of strategy will be required in the classroom. in general, there are few fully worked-out (and walked-through) examples of, e.g., truth-tables and proofs in the body of the text. Instructors will have to supply their own. No examples of translations of sentences with multiple connectives are given in the body of the text. Occasionally the text is arguably too brief. The description of Aristotelian logic in Chapter 1 strikes me as too condensed to be informative. I would like to have seen more discussion of the inevitability (given the constraints of bivalence and the definitions of other operators) of considering material conditionals with false antecedents to be true. The presentation, in section 3.4, of the "partial" truth-table method lacks consideration of its ability to discover facts (about validity, equivalence, etc.); instead, it's presented as a way merely of confirming what is already known (that an argument is invalid, e.g.). And in Chapter 4, difficult cases in quantificational logic are often passed over very rapidly: it is simply asserted that 'If anyone can play guitar, then Lemmy can' should be paraphrased such that the antecedent is existential (why not universal?); later, it is simply asserted, with no explanation, that "?xGx ? Gl means the same thing as ?x(Gx ? Gl), and ?x(Gx ? Gl) means the same thing as ?xGx ? Gl." Those are not intuitive equivalences. The book also moves very quickly. No practice on computing the truth-values of compound sentences under a single truth-value assignment is given before moving to full truth-tables. Multiply quantified sentences get really hairy, very fast--culminating with 'There is someone who likes everyone who likes everyone that he likes.' This could be a virtue or a vice, depending on the aptitude of one's students. The fifth chapter, on formal semantics, is in my view only suitable for high-aptitude students. It presents full model-theoretic semantics, with all the Tarkian bells and whistles. I do not present this material in my introductory logic course. Other difficult material--sections on ambiguous predicates, empty terms, Russell's theory of descriptions--is also unusual for an introductory course. But its inclusion meets the author's stated aim for the text, viz. to give students the ability to "be able to understand most quantified expressions that arise in their philosophical reading." This is a very good book if one's students are philosophy majors. The sections mentioned, and the structure and speed of the book, are appropriate for that audience. Some sections are exemplary: 6.8 beautifully shows how proof theory and formal semantics complement one another, and nicely sets the stage for proofs of soundness and completeness. However, if one's introductory logic class is populated mainly with non-majors (as ours is; it fulfills a university-wide formal reasoning requirement), then this book is not perfectly suitable. That said: the price is right! Properly supplemented, this book could be used for any introductory logic course.

## Table of Contents

- Chapter 1: What is logic?
- Chapter 2: Sentential logic
- Chapter 3: Truth tables
- Chapter 4: Quantified logic
- Chapter 5: Formal semantics
- Chapter 6: Proofs

## About the Book

*forall x* is an introduction to sentential logic and first-order predicate logic with identity, logical systems that significantly influenced twentieth-century analytic philosophy. After working through the material in this book, a student should be able to understand most quantified expressions that arise in their philosophical reading.

This books treats symbolization, formal semantics, and proof theory for each language. The discussion of formal semantics is more direct than in many introductory texts. Although forall x does not contain proofs of soundness and completeness, it lays the groundwork for understanding why these are things that need to be proven.

Throughout the book, I have tried to highlight the choices involved in developing sentential and predicate logic. Students should realize that these two are not the only possible formal languages. In translating to a formal language, we simplify and profit in clarity. The simplification comes at a cost, and different formal languages are suited to translating different parts of natural language.

The book is designed to provide a semester's worth of material for an introductory college course. It would be possible to use the book only for sentential logic, by skipping chapters 4-5 and parts of chapter 6.

## About the Contributors

### Author(s)

**P.D. Magnus** - In addition to loving wisdom, I am a philosopher by vocation. I am an associate professor at the University at Albany, State University of New York. I previously taught at UC San Diego (where I received my PhD) and at Bowdoin College.

My primary research is in the philosophy of science, motivated broadly by a falliblist but non-sceptical conception of scientific knowledge. I have written a lot on the underdetermination of theory by data, and my recent work is on natural kinds.