First-Order Logic (Dover Books on Mathematics) [Englisch] [Taschenbuch]

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.

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.