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Title: First-Order Logic by Raymond M. Smullyan ISBN: 0-486-68370-2 Publisher: Dover Publications Pub. Date: 01 February, 1995 Format: Paperback Volumes: 1 List Price(USD): $9.95 |
Average Customer Rating: 4.22 (9 reviews)
Rating: 4
Summary: Great stuff.
Comment: 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: 4
Summary: An Oddity But a Good-ity. Wait, that's terrible.
Comment: 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: 4
Summary: Wonderful--Why Can't I Assign It 4.5 Stars?
Comment: 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.
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Title: On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel ISBN: 0486669807 Publisher: Dover Publications Pub. Date: 01 February, 1992 List Price(USD): $6.95 |
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Title: The Riddle of Scheherazade: And Other Amazing Puzzles by Raymond Smullyan ISBN: 0156006065 Publisher: Harvest Books Pub. Date: 01 November, 1998 List Price(USD): $13.00 |
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Title: Godel's Proof by Ernest Nagel, James Roy Newman, Douglas R. Hofstadter ISBN: 0814758169 Publisher: New York University Press Pub. Date: 01 February, 2002 List Price(USD): $18.95 |
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Title: Computability and Unsolvability (McGraw-Hill Series in Information Processing and Computers.) by Martin Davis ISBN: 0486614719 Publisher: Dover Publications Pub. Date: 01 November, 1982 List Price(USD): $14.95 |
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Title: Introduction to Logic and to the Methodology of Deductive Sciences by Alfred Tarski ISBN: 048628462X Publisher: Dover Publications Pub. Date: 01 April, 1995 List Price(USD): $11.95 |
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