Mathematical Logic

Rating:
Summary: Very good but be aware of omissions
Review: This book is indeed much shorter than Principia, mainly because it is derived for lecture notes for a 1 semester PhD course. It is also a lot clearer than PM. But the notation is largely the same, which makes for hard reading if your are under 50. Quine's proof format doesn't take up much space, but has always eluded me. This book contains the best treatment of truth functional and quantificational logic prior to natural deduction and truth trees.

I like the set theory of this book, but I warn you that it is very nonstandard. Even ardent lovers of Quine's NF theory hate
the ML theory of this book.

The weakness of this book is its treatment of metatheory:
consistency, completeness, decidability, categoricity. The treatment of Godel's incompleteness is detailed and highly original (altho' it owes more to Tarski than to Godel). But it is very difficult, and Smullyan (1991) is much better.
Quine also had no clue re model theory or recursion.

I respect the historical remarks a lot. Just one big omission: Quine, like nearly everyone of his generation, missed that
math logic as we know and love it does not descend from Frege, but from an 1885 article by C S Peirce.

Rating:
Summary: In Depth Look at Logic
Review: Try this book when you know a bit about the basics of logic. The descriptions are much more lucid than those in Principia, even if the ideas are less earthshattering for there time. Quine, as he always does, gives a masterful, detailed look at logic. If you are a fan of logic and the foundations of math, this book is not to be missed.