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Logic for Mathematicians (Dover Books on Mathematics) Paperback – December 18, 2008
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In this Logic for Mathematicians Dover book by John Barkley Rosser, the Whitehead/Russell theory of types is replaced by "Quine's New Foundations" (page 206), which is an unfortunate choice, but nowhere near as bad as the Whitehead/Russell theory of types!
Rosser uses the almost unreadable Peano-style dot-notation instead of modern grouping-parentheses, but does use parentheses for logical quantifiers. So the set of non-negative integers is defined like this:
which I think most people would find difficult to parse! (There's actually a hat on the first "x" which I can't type in HTML.) By the way, that definition of the integers (if you do make the effort to parse it), says effectively that the integers are the intersection of all classes which contain zero and all of its successors. That's not the kind of definition which most working mathematicians would prefer to use these days.
Rosser gives an excellent presentation of the possible solutions to Russell's paradox on pages 197-207. There are many fascinating historical discussions, as well as lengthy comments on the application of logic to practical mathematics.
Pages 12-76 present a 3-axiom propositional calculus for the operators "⇒", "∧" and "¬", using modus ponens. Then there are presentations of predicate calculus with and without equality. Then classes are axiomatized. This is followed by relations and functions, cardinal numbers (111 pages: 345-455), ordinal numbers, the integers, and the axiom of choice. Unfortunately, this is all done in archaic notations in an old-fashioned style. This book has considerable historical interest, but it cannot be recommended as an introduction to modern logic.
My opinion of this book has increased recently. In particular, I am very happy with Rosser's treatment of predicate calculus. I am referring here particularly to "Rule C" on pages 126-149, which according to Margaris, First Order Mathematical Logic, page 191, was introduced by Rosser, who shows that any theorem which can be proved with Rule C can be proved without Rule C. This rule is a "choice rule", which is how every real mathematician uses existential quantifiers in proofs. And this shows the enormous strength of this book by Rosser. He presents logic in a way which is consistent with how mathematicians do mathematics. Most logic books before Rosser were using an equivalent of Rule C, called "existential quantifier elimination", or EE for short. But after Rosser, this rule has been omitted by most logic textbooks. So this book by Rosser seems to mark the point in history that predicate calculus treatments abandoned the natural way in which mathematicians really work.
And just one other point about the dot-notation which Rosser uses. The best way to interpret it is to snip the expression where the largest set of dots occurs, and put parentheses around both halves. Then do this recursively until you have a parenthesized expression.
Another really good thing about the Rosser book is that it gives applications to the foundations of mathematics in a way which is useful to mathematicians, following the topic sequence of the Whitehead and Russell 3-volume Principia Mathematica. Most other logic textbooks roam off into proof theory and model theory and extreme abstractions which have little relevance to mathematicians. So if you want logic which is useful for doing mathematics, this might be the right book for you.