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First-Order Logic (Dover Books on Mathematics)

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Raymond M. Smullyan (Author)
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Raymond M. Smullyan (Author)
4.2 out of 5 stars 12 customer reviews
ISBN-13: 080-0759683703
ISBN-10: 0486683702
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Product Details

  • Series: Dover Books on Mathematics
  • Paperback: 158 pages
  • Publisher: Dover Publications (January 30, 1995)
  • Language: English
  • ISBN-10: 0486683702
  • ISBN-13: 978-0486683706
  • Product Dimensions: 0.5 x 5.5 x 8.5 inches
  • Shipping Weight: 2.4 ounces (View shipping rates and policies)
  • Average Customer Review: 4.2 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #208,583 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

65 of 67 people found the following review helpful
A Stripped-Down Exposition of a Bare-bones Subject
By James R. Mccall on June 15, 2001
Format: Paperback Verified Purchase
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.Read more ›
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40 of 41 people found the following review helpful
Great stuff.
By Jason T on March 2, 2004
Format: Paperback
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.
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23 of 24 people found the following review helpful
a classic
By Todd Ebert on February 15, 2002
Format: Paperback Verified Purchase
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.
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24 of 28 people found the following review helpful
An Oddity But a Good-ity. Wait, that's terrible.
By J. Wiley on January 29, 2003
Format: Paperback
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.
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Most Recent Customer Reviews

Five Stars

Great!

Published 13 months ago by bdonaat
Three Stars

great book but it does speak over my head

Published 16 months ago by MD90
Dense, clear, factual... A thrill !!!

... A thrill if you can cope with the high-speed, no fuss, complete but concise style, i.e. Smullyan's way at its best ! Read more

Published on February 28, 2012 by André Gargoura
Great as a Reference, Probably Not for True Beginners

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. Read more

Published on December 12, 2001 by Dino
Too Concise

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... Read more

Published on January 31, 2001 by D. S. Heersink
The best you can get.

Smullyans "First-Order Logic" is the best of its kind. Easy to read, well-writen and with yust the nesesary formalism. Read more

Published on March 25, 2000
Don't read this review until you've bought this book!

When I first bought Smullyan's book 'First Order Logic : Ergebnisse De Mathematik Und Ihrer Grenzgebiete; Vol 43', I thought all of those words after 'First order Logic:' were big... Read more

Published on October 5, 1999
Very concise, yet crystal clear (no prerequisites!)

Sure, one would like Smullyan to elaborate a bit on the significance of this or that result. But if you want "just the facts (on FOL), please" this is the book for you. Read more

Published on June 9, 1999

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