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Logic and Structure 4th Edition
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This book is an introductory treatment of mathematical logic, written entirely from the perspective of GENTZEN natural deduction. Standard texts are written from the viewpoint of Hilbert axiomatics or from (an alternative from Gentzen) natural deduction. Thus, for one wanting any discussion of the Gentzen calculus ND this book is the only full, start from scratch, treatment that I know of. Note, however, that the Gentzen sequent calculus is NOT discussed.
Another (great) feature of the text is the chapter on ND in intuitionistic systems. Again, other than texts on structural proof theory (e.g. Negri and Von Plato.), such discussions are hard to find in an introductory setting. It also beats digging through journals or symposium proceedings.
Also, another (great) component, found only in the 4th edition, is the treatment of Godel's first famous result, but entirely handled via the aforementioned Gentzen calculus.
Finally, to give a general comment, there are reasons for wanting to treat Hilbert systems rather than the Gentzen systems; these are most easily extended to study theories of arithmetic. But from the strictly logical, proof theoretic aspects, Hilbert systems prove to be less appealing. Note however that Gentzen systems are being pushed further all the time to handle arithmetic theories, though these still might be less elegant than their counterparts. Anyway, whether you choose Gentzen or the more standard treatment of Hilbert systems, one will unavoidably be making some concessions; but if Gentzen is what you want, there really isn't another alternative than this. Anecdote: Sometimes I hear, Gentzen is the more pedagogically effective route, but I'm not sure about this.
More fundamentally, Van Dalen skips a lot of necessary explanations and some of his statements are ambiguous, confused, confusing as he attempts to relieve the reader from the difficulties of over-formalization found in other authors of logic indigestions such as Enderton and Mendelson : so we fall from Charybdis into Scylla !!!
I stopped losing my time with this book and turned to Kleene's beautiful and masterful achievements "Introduction to Metamathematics" and "Mathemathical Logic" (see my reviews).
Fortunately, there are other books on the subject ! Here is a list of key books :
1. A. Tarski's "Introduction to Logic", a jewel, followed by P. Smith's superb entry-point "An introduction to Formal logic" and the lovely "Logic, a very short introduction" by Graham Priest
2. D. Goldrei's "Propositional and Predicate calculus"
3. Wilfrid Hodges' "Logic", followed by Smullyan's "First-order logic".
4. P. Smith's "An introduction to Gödel's theorems".
5. Kleene's "Introduction to metamathematics" & "Mathematical Logic".
6. G. Priest's " Introduction to non-classical logic".
Hence forgetting altogether Van Dalen's indigestible "Logic & Stucture" as well as
the even more indigestible Enderton, Mendelson & al...