First-Order Logic

Rating:
Summary: Great stuff.
Review: First, this isn't one of Smullyan's popular puzzle books- its a serious mathematics text. Second, don't use this as your first exposure to first-order logic (note the title doesnt say "Introduction to ...")- although logically self-contained, it requires some experience to appreciate what a neat little book this is.

It's not a general mathematical logic text- there is no model theory (beyond basic Skolem-Lowenheim), incompleteness, recursion theory, or set theory. It covers tableaux (this alone is worth the price of the book), Hilbert-style axiomatic systems (briefly), sequent systems, Gentzen's Hauptsatz and Extended Hauptsatz, Craig's and Beth's theorems, and more. But the heart of the book is completeness theorems, their proofs, and closely related material such as compactness and Herbrand-like theorems. Smullyan shows there are two main approaches to completeness (analytic vs. synthetic), breaks each into stages, provides nice abstracted formulations, and usually gives several different proofs of each result. The centerpiece is his "Fundamental Theorem of Quantification Theory", a theorem associating a truth-table tautology with every valid first-order sentence (check out the amazingly slick proof of completeness for the the Hilbert-style system that this provides). Similar constructions such as magic sets are also discussed. All this forms a much more extensive and illuminating look at completeness proofs than I've seen elsewhere.

The first-order logic used in the book has no equality and no function signs. There are few exercises, most of them simple. Smullyan writes clearly and with an appropriate amount of rigor (but its not as polished as his later books). Makes a great supplement to more general-purpose introductory mathematical logic books. If you haven't seen the tableau method yet buy this book immediately. Experienced readers will appreciate the sophisticated coverage of completeness proofs.

Rating:
Summary: Great stuff.
Review: First, this isn't one of Smullyan's popular puzzle books- its a serious mathematics text. Second, don't use this as your first exposure to first-order logic (note the title doesnt say "Introduction to ...")- although logically self-contained, it requires some experience to appreciate what a neat little book this is.

It's not a general mathematical logic text- there is no model theory (beyond basic Skolem-Lowenheim), incompleteness, recursion theory, or set theory. It covers tableaux (this alone is worth the price of the book), Hilbert-style axiomatic systems (briefly), sequent systems, Gentzen's Hauptsatz and Extended Hauptsatz, Craig's and Beth's theorems, and more. But the heart of the book is completeness theorems, their proofs, and closely related material such as compactness and Herbrand-like theorems. Smullyan shows there are two main approaches to completeness (analytic vs. synthetic), breaks each into stages, provides nice abstracted formulations, and usually gives several different proofs of each result. The centerpiece is his "Fundamental Theorem of Quantification Theory", a theorem associating a truth-table tautology with every valid first-order sentence (check out the amazingly slick proof of completeness for the the Hilbert-style system that this provides). Similar constructions such as magic sets are also discussed. All this forms a much more extensive and illuminating look at completeness proofs than I've seen elsewhere.

The first-order logic used in the book has no equality and no function signs. There are few exercises, most of them simple. Smullyan writes clearly and with an appropriate amount of rigor (but its not as polished as his later books). Makes a great supplement to more general-purpose introductory mathematical logic books. If you haven't seen the tableau method yet buy this book immediately. Experienced readers will appreciate the sophisticated coverage of completeness proofs.

Rating:
Summary: a classic
Review: I mainly bought this book because of the influence it has had on numerous modern-day logic texts. If you are unfamiliar with the tableaux method for structural proofs, then you will gain alot from reading this, as it provides a different perspective from the more popular Hilbert-system approach. Tableaux systems, of course, have been made popular because they are easy to program with a computer. Please see Gallier's "Logic for Computer Scientists" for more on this matter.

Rating:
Summary: An Oddity But a Good-ity. Wait, that's terrible.
Review: The reviewer from Illinois gave a very good characterization of Smullyan's style here:
"Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof."
Readers familiar with Smullyan's enormous talent for popular exposition may be expecting the same herein: not so. This is very much for people who have attained what medical professionals call "mathematical maturity" (which is about as difficult to attain as zen, yet perhaps amounts to little more than the ability to read VCR instruction manuals). For example, the very first section is a wiz-bang treatment of trees (not the usual graph-theoretic ones), defined in the abstract/axiomatic fashion.
Of course, people who spend perhaps way too much of their time steeped in math are attracted to treatments of just this sort.
A structural characterization in terms of sets and mappings is much more meaningful, interesting, and aesthetically pleasing to those with these unusual inclinations (compulsions?) than a characterization framed significantly by historical motivation (please understand that I'm speaking roughly here). This is why I gave a positive review. A star was witheld for the selfish reason that I'm not sure I'll find much use for such an odd treatment of model theory, the topic for which I was seeking a more mainstream treatment when I purchased this. Regrets are nonetheless few: time spent reading Smullyan is never a waste.

Rating:
Summary: An Oddity But a Good-ity. Wait, that's terrible.
Review: The reviewer from Illinois gave a very good characterization of Smullyan's style here:
"Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof."
Readers familiar with Smullyan's enormous talent for popular exposition may be expecting the same herein: not so. This is very much for people who have attained what medical professionals call "mathematical maturity" (which is about as difficult to attain as zen, yet perhaps amounts to little more than the ability to read VCR instruction manuals). For example, the very first section is a wiz-bang treatment of trees (not the usual graph-theoretic ones), defined in the abstract/axiomatic fashion.
Of course, people who spend perhaps way too much of their time steeped in math are attracted to treatments of just this sort.
A structural characterization in terms of sets and mappings is much more meaningful, interesting, and aesthetically pleasing to those with these unusual inclinations (compulsions?) than a characterization framed significantly by historical motivation (please understand that I'm speaking roughly here). This is why I gave a positive review. A star was witheld for the selfish reason that I'm not sure I'll find much use for such an odd treatment of model theory, the topic for which I was seeking a more mainstream treatment when I purchased this. Regrets are nonetheless few: time spent reading Smullyan is never a waste.

Rating:
Summary: Too Concise
Review: This book covers a lot of territory with very few words, and most of those words are so unexplanatory that, if one doesn't have a good understanding of formal logic aforehand, he or she will be lost in the brevity. Brevity generally facilitates clarity, but here the loss of language to hang one's hat on leaves the reader looking at symbols without sufficient reference to give them meaning, much less answer the question, Why? This is NOT an introductory text, which may be implied by the phrase "first-order" in the title. This is boolean and mathematical throughout.

Rating:
Summary: A Stripped-Down Exposition of a Bare-bones Subject
Review: This is a book by a man I knew for his books of puzzles-chatty books of great originality that have fun with the paradoxical possibilities of logic. Here he is the teacher of logic, and aside from an occasional phrase, the serious mathematician. However, Smullyan's originality shines through in this book as well. He presents logic as a branch of mathematics rather than an abstraction of ordinary language. And he uses a method from the recent literature, tableaux, to build his proofs in a simple and satisfying way. He gets directly to the main result as to the provability of valid sentences using this method for both the propositional calculus and the predicate calculus.

Smullyan procedes rapidly because he makes some assumptions about the reader's knowledge. The reader must understand the difference between mathematics and meta-mathematics-that is, should be able to separate out the talking about the sentences of the system, which may contain (among other signs) the conjunction, disjunction, and negation, from the more-or-less informal arguments that prove assertions about these sentences using natural language, with its "and", "or", and "not". Moreover, the concept of "proof" is used at two levels: the particular tableau that constitutes a proof of a sentence, and the "proofs" about tableaux and other concepts of the "system".

Besides this, the reader should have a good feel for recursive definitions, which are used everywhere. Finally, this model reader should know the difference between countably-infinite sets and uncountably-infinite sets.

I knew all that, but still found the text slow going, maybe because I have been away from mathematics for decades. But there is another reason, too. Smullyan has divorced logic from its roots: logics are simply recursively-defined sets of sentences and mappings, and that is that. No discussions, ala WvO Quine, on the history or linguistic difficulties of a concept, just definition and proof. This is an abstraction of a subject which is already an abstraction. So I usually found myself trying to understand what it all meant, in other than these stark set-and-mapping terms. On the other hand, many difficulties caused by the details of historical development of the subject vanish, and the results stand-... simple, directly derived.

This is a slender Dover volume, of high quality and low cost. I would have given the book 5 stars, but for two things. The exercises are too hard, sometimes, and without answers, and the index is very poor. Still, I think the treatment is the best around for those who want to use logic as a basis for studying incompleteness or proof theory. It is not to be confused with a more full-blown treatment that also treats logic as a branch of the humanities.

Rating:
Summary: Wonderful--Why Can't I Assign It 4.5 Stars?
Review: This is a great book, and served as my introduction to tableaus. I think it strikes a good balance between the conciseness of a math text and the verbosity often found in philosophy texts; it's also very reasonably priced. My only complaint is one that I noticed in a previous review: some exercises are too difficult and there are no solutions (the former wouldn't be a problem if the latter weren't the case). Also, this book isn't for the total rookie-some prior knowledge is assumed. My choice for introductory material would be Copi's Symbolic Logic or even his Introduction to Logic (with Cohen) for those with no or limited background in mathematics.

Rating:
Summary: Great as a Reference, Probably Not for True Beginners
Review: This is an excellent reference! It has more material covered in just 155 pages than most other works address in twice as much space. I refer to it very often.

However, I doubt it would be appropriate for someone that has not previously been introduced to the material. If a truly introductory text is required, I would look elsewhere.