There are some things not explicit in the posted summary that buyers of this book should know. Researchers will also find these comments valuable
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 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.