Customer Reviews

Metalogic: An Introduction to the Metatheory of Standard First Order Logic

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I must respectfully disagree with many of the negative things said in the previous reviews about this book. I find it to be by far and away the easiest to understand text on metalogic that I have come across. I am, however, a professional philosopher with a good deal of training in logic. So the material is not new to me, which surely makes the book much easier for me to understand. That said, I think that some of the previous reviewers have unfairly judged this book owing to the complexity of the material it covers. One should remember that Hunter's aim is not to present rigorous arguments but to INTRODUCE metalogic. I am sure he would be satisfied if his reader is inspired to look into any of the proofs in more detail elsewhere. As far as the claim that the book contains errors, I have yet to read a logic text that didn't!

I feel compelled to write this review because I disagree utterly with the previous reviewer. This is a fantastic book! I have no specialist training in logic, but I found "Metalogic" to be a clear, interesting, well-written and definitely "painless" text. I highly recommend this book as an introduction to metalogic; in fact, I would recommend it in preference to the other popular text on metalogic, the brilliant "Godel Escher Bach". Whereas "Godel Escher Bach" introduces the field of metalogic using allegories, parables, illustrations and copious references to turtles, "Metalogic" introduces it using clear, simple arguments and gentle, step-by-step explanations.

A quote from the preface: "Many elementary logic books stop just where the subject gets interesting. This book starts at that point and goes through the interesting parts...." Many of us who have struggled through the dry and boring foundations of elementary logic have become aware that not all logic is as tedious as the foundations. Beyond elementary logic lies a fascinating world inhabited by the paradoxes of Bertrand Russell, the counterintuitive proofs of Turing, Church and Godel, and the mindboggling infinite sets of Cantor. The problem is, just where logic becomes interesting, it also becomes unattainable to the average Joe. "Metalogic" undertakes to introduces this field to those without the specialist training that is normally required, and it succeeds admirably.

I have a dozen books on Foundations, some specifically on Logic, some Logic with Set Theory (e.g., Tourlakis), some pure Set Theory. This one is one of my favorites. I am an accomplished mathematics reader, and more than occasional doer, even though not a mathematician per se (Electrical Engineering is my field). My experience with the subject of this book has not been that of the professional philosopher or logician, but I have endeavored to read and absorb the material in all of my sources. This book is easily the one that opens the door most cordially, assuming little (which I appreciate since I am not getting a professor's help with subtleties nor that priceless mnemonic support that comes from classroom gestalts induced by a great expositor). The read somehow seems more perspicuous than the others, allowing the reader to feel more confident. The fact that Hunter takes the reader from first principles through decidability without holes or hand-waving is remarkable, and one suffers a minimal amount of that nagging feeling that something wasn't fully understood. (Of course, that feeling is normal when breaking in new ideas, but it's nice when you can go pages between bouts.) I recommend the book highly. When I thought I had lost it (I have almost 1000 books) I was ready to buy a replacement. As in all studies, the reader does well to use multiple sources, so he need not make this the "one and only". But I'd surely start with it.

I didn't have a choice in this logic book as it was prescribed by my logic professor. Difficult to understand sure, but only if you aren't paying close attention. Errors, there are always errors in textbooks, it only matters if they are fundamental to understanding the content. Without the proper training in this discipline I would suggest you not even try, it is much like another language, in fact it is another language, a metalanguage. About halfway through the class and it is not as bad as other reviewers have stated.

I understand this is an introductory book, although my true introduction was symbolic logic and calculus. These two classes will most definitely help in comprehending this book.

I found this book terribly disappointing. Contrary to what the reviews in the backcover say, it is not at all "painless" to read nor does Hunter provide "numerous excercises". Excercises are indeed scarce (less anyway than what many standard texts introducing mathematical logic include); as regards painfulness, the reasoning in the book is at crucial points enthymematic, something highly disturbing for the beginner. The book also contains mistakes. For instance, on page 153, after the definitions of satisfaction and satisfiability, we read: "VvA is satisfiable for an interpretation I iff A is satisfiable for the same interpretation". But no meaning was previously attached to the expression "satisfiable for interpretation I", but only the usual definitions of "satisfaction for interpretation I" and "satisfiable" (simpliciter). Another example: there is a mistake in case 3 of the induction step of the proof of theorem 40.12, p. 156: the phrase that begins in the first line and ends in the third states something true of A but not relevant: what we need to prove is that the same holds for B, in order to apply the inductive hypothesis. Also, while the Basis of the proof of the same theorem should have employed induction on the number of function symbols, it just appeals to an "intuitive" explanation. There are also mistakes and misprints on page 199, in the proof-sketch of theorem 47.2. And at some other places as well.
The very same sketchy (or cursory) character of the whole book makes it unsuitable for the beginner and unhelpful for the specialist.

This book gives examples and "answers" to the examples. It makes the clear distinction between formality or "syntax" and meaning or "semantics." This critical distinction is utilized to shed light on consistency proofs and undecidability. Part of Goedel's proof involved the liar's paradox. In other words, part of Goedel's proof required the semantic acknowledgement of a phrase that didn't go anywhere, namely, "this theorem is not provable" which, as model for the formal sentence or syntax "S", established proof that the formal sentence couldn't go anywhere. How tough is that? Goedel's proof is overrated for its complexity if one takes an inventory of the aforementioned simple fact which is the core of his argument. In other words, to establish Goedel's proof, in lieu of the tradition bequeathed to us from generations of mathematicians, such as Pappus' proof (best illustrated in D'Abro's book on

"The Rise of the New Physics"), we need to take a step in the the semantic world and then take a step in the formal/syntax world. All proofs are dependent on the interchange of semantics and syntax. Some things are easier to solve in the syntax world. If not, some things are easier to solve in the semantic world. If a semantic argument is easy, such as the liar paradox, we acknowlege its mirror image in the syntax world, and by solving something in the semantic world we know we've accomplished something in the syntax world, and vice versa. That's all.

I'm sure this book was "readable" and "painless" to people who have degrees in the field, but I found it unnecessarily dry and obscure. Hunter may be a good logician (it's hard for me to say, since after taking the course and reading the book I still have no grasp of the subject), but he has no facility with getting his point across concisely and understandably. If you're a professor considering this book as a text, unless you are HIGHLY competent, nay, EXCELLENT teacher, please reconsider. I must admit that some of my negative opinion about Hunter does come from having a lousy professor and thus a generally miserable time with the material, but if the text had been "readable" and "painless" and "an excellent introduction" I would have been able to learn the stuff myself.

Hegel said: "Impatience tries the impossible: to attain the end without the means". Well, this book is Impatience in metalogical affairs: it procureth to explain first order meta-logic without what is required: detailed and rigurous examination.

Let me disagree whith the reader who disagrees with the reviewer who calls Hunter's book cursory. I think the mistakes to which this last reviewer points are precisely the result of Hunter's cursory and rash style. I hadn't noticed the one on page 156.

The book begins nicely: it explains quite well some elementary matters of set theory. But even here sequences (very important things one has to deal with while doing logic) are wrongly taken to be n-tuples. There is trouble when he explains the length of a sequence: the "terms" of a sequence are taken to be the "objects ordered by the sequence"; the reader might then be tempted to think that a sequence ABCA orders 3 objects, and therefore is a sequence of 3 terms (while it is a sequence of 4 terms); the denumerable sequence AAAAA.... might be thought to be a sequence of 1 term, namely A. If Hunter had taken the trouble to define a sequence, and the "nth term of a sequence", in the usual manner (as a function f from an initial segment of natural numbers, and as f(n) respectively), these inconveniences could have been most naturally avoided. It is true that the reader will look at Hunter's own examples and probably figure out how things really work. But he will be entirely justified if he complains about the ambiguous explanations of the author, especially since Hunter KNOWS that sequences should be defined as functions (see p.25).

The rest of the book is clear and legible in some parts, but hasty in others. It is quite clear when it deals with truth functional logic, but hasty when it comes to consistency of first order logic, and over-hasty when it explains undecidability. It would be boring (for me) to point to the details right now.

Given that there are books which explain the whole subject with greater clarity and precision, I would not recommend this book to any serious reader or buyer.